## 5 Signal Approximation and Estimation
### Combination of ARXU and MBR for Signal Approximation
Moreover, based on Equation (23), the integral term is selected to reduce the effect of uncertainties in signal estimation. To improve the accuracy of the AMRXU-PI observer, the AMRXU-PMI observer is recommended here. Therefore, based on Figure 1, to estimate the resampled RMS signal for healthy and faulty conditions, and to improve the accuracy of the AMRXU-PI observer, in the second step, the AMRXU-PMI observer is selected, and the state-space equations are as follows:  
\[\begin{array}{ll}X_{AMRXU-PMI}(k+1)&=\ [\ell_{X}X_{AMRXU-PMI}(k)+\ell_{ \epsilon}\epsilon_{AMRXU-PMI}(k)+\ell_{MBR}h_{MBR}]\\ &\ \  
Moreover, to minimize \(UIP_{SNFS}\) based on gradient descent, we have  
\[ \begin{array}{rl}(UIP_{SNFS})_{min}&=\frac{\partial}{\partial\mathcal{C}_{k}} \times[\frac{k}{1}(\epsilon_{AMRXU-VSPMI}(k)-\epsilon_{AMRXU})\times\gamma_{k} ]^{2}+\frac{\partial}{\partial\gamma_{k}}\\ &\times[\frac{k}{1}(\epsilon_{AMRXU-VSPMI}(k)-\epsilon_{AMRXU})]^{2}\frac{ \partial\gamma_{k}}{\partial\mathcal{C}_{k}},\\ &=2(\epsilon_{AMRXU-VSPMI}(k)-\epsilon_{AMRXU})\\ &\times\gamma_{k}(\epsilon_{AMRXU-VSPMI}(k)-\epsilon_{AMRXU})\times\gamma_{k }\times\frac{\partial\gamma_{k}}{\partial\mathcal{C}_{k}}.\end{array}  \tag{36}\]  
Regarding Equation (36), by updating \(C_{r}\) and \(\sigma\), we can minimize \(UIP_{SNFS}\). That means the performance from TSK fuzzy logic improves. Therefore, the adaptive \(C_{k}\) and \(\sigma\) are represented as follows [44]:  
\[\mathcal{C}_{k,t+1}=\mathcal{C}_{k,t}-\varnothing_{k,t}\frac{\partial UIP_{ SNFS}}{\partial\mathcal{C}_{k,t}}. \tag{37}\]  
and  
\[\sigma_{k,t+1}=\sigma_{k,t}-\varnothing_{k,t}\frac{\partial UIP_{ SNFS}}{\partial\sigma_{k,t}}. \tag{38}\]  
Here, \(\varnothing_{k,t}\) is the tuning coefficient. Therefore, based on Figure 1, to improve the flexibility and accuracy of signal estimation in the presence of uncertainties in the AMRXU-VSPMI observer, in the last step, the AMRXU-AVSPMI observer is selected, and the state-space equations are as follows:  
\[ \begin{array}{rl}X_{AMRXU-AVSPMI}(k+1)&=[\ell_{X}X_{AMRXU-AVSPMI}(k)+\ell_{ \epsilon}\epsilon_{AMRXU-AVSPMI}(k)+\ell_{MBR}h_{MBR}]\\ &+\ell_{\epsilon}\epsilon_{AMRXU-AVSPMI}(k)+\epsilon_{AMRXU-AVSPMI}\\ &+\ell_{\rho}(Y_{AMRXU-AVSPMI}(k)-Y_{AMRXU}(k))+[X_{AMRXU-AVSPMI}(k)\\ &+\ell_{\ell_{2}}()X_{AMRXU-AVSPMI}(k)-X_{AMRXU}(k))]\\ \end{array}  \tag{39}\]  
\[ \begin{array}{rl}Y_{AMRXU-AVSPMI}(k)=\ell_{Y}{}^{T}X_{AMRXU-AVSPMI}(k),\\ \\ \epsilon_{AMRXU-AVSPMI}(k)=\gamma_{u,rms}(k)-Y_{AMRXU-AVSPMI}(k-1)\\ \\ \epsilon_{AMRXU-AVSPMI}(k+1)=\epsilon_{AMRXU-AVSPMI}(k)+\ell_{i(1)}(\epsilon_{ AMRXU-AVSPMI}(k)-\epsilon_{AMRXU}(k))\\ +\ell_{VS}\times sgn\parallel(\epsilon_{AMRXU-AVSPMI}(k)-\epsilon_{AMRXU}(k)) \parallel+\ell_{SNFS}\epsilon_{SNFS}(k).\end{array}  \tag{40}\]  
Here, \(X_{AMRXU-AVSPMI}(k)\), \(Y_{AMRXU-AVSPMI}(k),\epsilon_{AMRXU-AVSPMI}(k),\epsilon_{AMRXU-AVSPMI}\), and \((\ell_{SNFS})\), respectively, are the state estimation of RMS signal modeling, the output estimation of RMS signal modeling, the estimation error, and the uncertainties estimation (all using the AMRXU-AVSPMI observation technique), plus the state-space tuning coefficients for the SNFS term. Based on Equation (42), the estimation of uncertainty using the SNFS term increases accuracy and flexibility in the uncertainty estimation of RMS signal modeling.