# Fault Estimation for Semi-Markov Jump Neural Networks Based on the Extended State Method
## 2 Preliminaries  
The following class of discrete-time s-MJNNs with time-varying delay is considered in the below form:  
\[ \begin{array}{ll}x(g+\ 1)&=&A(r(g))x(g)\ +B(r(g))f(x(g))\ +C(r(g))f(x(g-d(g)))\\ &&+D(r(g))\omega(g)+H(r(g))fa(g)\\ y(g)&=&E(r(g))x(g)+G(r(g))fs(g)\\ z(g)&=&F(r(g))x(g))\\ x(g)&=&\overline{x}(g),g=-d_{2},-d_{2}+1,\ldots,0,\end{array}  \tag{1}\]  
where \(fa(g)\) is the actuator fault; \(fs(g)\) is the sensor fault; \(x(g)\in R^{l}\) and \(y(g)\in R^{n}\) represent the neural state vector and the measurement output, respectively; \(z(g)\in R^{m}\) is the estimated output signal, and \(\omega(g)\in R^{w}\) denotes an external disturbance constrained within the range \([0,\infty]\). \(A(r(g))=diag(a_{1}(r(g)),\ldots,a_{n}(r(g)))\) is the positive diagonal matrix. \(B(r(g))\), \(C(r(g))\), and \(D(r(g))\) represent the known weight matrices. \(R^{l}\), \(R^{n}\), and \(R^{w}\) denote the \(l\)-dimensional, \(n\)-dimensional, and \(w\)-dimensional real vector spaces, respectively. \(f(x(g))=[f_{1}(x_{1}(g)),f_{2}(x_{2}(g)),\)\(f_{3}(x_{3}(g)),\ldots,f_{n}x_{n}(g)]^{T}\in R^{n}\) is the neural activation function. \(E(r(g))\), \(F(r(g))\) is the interconnection weight matrices. \(H(r(g))\) represents the fault influence matrix, which characterizes how actuator faults affect the system state. \(G(r(g))\) is the sensor fault influence matrix, describing how sensor faults alter the measured output. \(d(g)\) is marked by the time-varying delay and satisfies \(d_{1}\leq(g)\leq d_{2}\), where \(d_{1}\) and \(d_{2}\) are the upper and lower bounds of the time delay. The parameter \(r(g),g>0\) represents a semi-Markov chain, whose evolution is governed as follows:  
\[P\{r(g+\zeta)=q|r(g)=p\}=\left\{ \begin{array}{l}\pi_{pq}(\zeta)\chi+o(\zeta ),\ if\ p\
eq q,\\ 1+\pi_{pp}(\zeta)\chi+o(\zeta),if\ p=q,\end{array} \right\}\]  
where \(\zeta>0\) is the sojourn time. \(lim_{\zeta\to 0}(o(\zeta)/\zeta)=0\) and \(\pi_{pq}(\zeta)\geq 0\) for \(p\
eq q\) are the transition rates from mode \(i\) at time \(g\) to mode \(q\) at time \(g+\chi\) and \(\pi_{pq}(\chi)=-\sum_{q=1,q=p}^{Q}\pi_{pq}(\chi)\). **Remark 1**. : _This study considers that faults and disturbances within the system are distinct, with disturbances being constrained within certain bounds._Considering the limited bandwidth of communication channels from sensors to the observer, data congestion and other similar issues may arise. To address these challenges, this paper introduces the RRP approach to alleviate network congestion. Let \(\widetilde{y}_{\ell}(g)=(\ell\in 1,2,3,\ldots,m)\) represent the measurement output received from the \(\ell\)th sensor. Following the RRP, \(\widetilde{y}_{\ell}(g)\) can take the following form:  
\[\widetilde{y}_{\ell}(g)= \begin{cases}y_{\ell}(g),&\text{if }\text{mod}(g-\ell,\ m)=0,\\ \widetilde{y}_{\ell}(g-1),&\text{other}.\end{cases}  \tag{2}\]  
In accordance with (2), it can be reformulated as follows:  
\[\widetilde{y}(g)=\psi_{\sigma}(g)y(g)+(I-\psi_{\sigma}(g))\widetilde{y}(g-1), \tag{3}\]  
where \(y(g)=\left[y_{1}{}^{T}(g),y_{2}{}^{T}(g),\ldots,y_{m}{}^{T}(g)\right]^{T}\), \(\widetilde{y}(g)=\left[\widetilde{y}_{1}^{T}(g),\widetilde{y}_{2}^{T}(g), \ldots,\widetilde{y}_{m}^{T}(g)\right]^{T}\), \(\sigma(g)=mod(g-\ell,m)+1\), and \(\psi_{\sigma(g)}=diag\{\tau(\sigma(g)-1)I,\tau(\sigma(g)-2)I,\ldots,\tau( \sigma(g)-m)I\}\) with \(\tau(\cdot)\in\{0,1\}\) is the Kronecker delta function. To simplify the notation, \(r(g)\) is represented by \(p\).