## 5 Signal Approximation and Estimation
### Combination of ARXU and MBR for Signal Approximation
In addition, bias is determined using the following function:  
\[\Theta=\frac{1}{s}\sum_{s\in S}\Bigg{[}Y_{s}-\sum_{i\in S}(\partial_{i}{}^{+} -\partial_{i}{}^{-})\times(\theta_{i},\theta_{\text{s}})-\beta\times sgn\big{(} \partial_{i}{}^{+}-\partial_{i}{}^{-}\big{)}\Bigg{]}. \tag{14}\]  
Here, \(Y_{s}\) and \(S\) denote the support vector resampled RMS signal in a normal condition, and the support vector, respectively. The support vector is defined as  
\[S=\big{\{}i\big{|}0\leq\partial_{i}{}^{+}-\partial_{i}{}^{-}\leq\Delta\big{\}}. \tag{15}\]  
Here, \(\Delta\) is a constant. So, the combination of the ARXU and MBR algorithms (or AMRXU) is represented with the following definitions:  
\[X_{AMRXU}(k+1)=[\ell_{X}X_{AMRXU}(k)+\ell_{\varepsilon}\epsilon_{AMRXU}(k)+ \ell_{MBR}h_{MBR}]+\ell_{\varepsilon}e_{AMRXU}(k), \tag{16}\]  
\[Y_{AMRXU}(k)=\ell_{Y}{}^{T}X_{AMRXU}(k), \tag{17}\]  
\[e_{AMRXU}(k)=Y_{n,rms}(k)-Y_{AMRXU}(k-1), \tag{18}\]  
\[\varepsilon_{AMRXU}(k)=Y_{AMRXU}(k)-Y_{n,rms}(k). \tag{19}\]  
Here, \(X_{AMRXU}(k),\ Y_{AMRXU}(k),\varepsilon_{AMRXU}\), and \(\ell_{MBR}\), respectively, are the state of RMS signal modeling using AMRXU, the output of RMS signal modeling using AMRXU, the uncertainties-effect estimation and feedback as inputs to reduce the effect of uncertainties of using AMRXU, and the state-space tuning coefficient. Combination of AMRXU-PMI Observer, VS-Lyapunov Algorithm, and SNFS Technique for Resampled RMS Signal Estimation  
According to Figure 1, after modeling the signal and extracting the nonlinear state-space equation from the resampled RMS signal in a healthy condition, normal and abnormal resampled RMS signals are estimated using the proposed self-tuning hybrid-based scheme. Although the accuracy of resampled RMS normal signal modeling using AMRXU is 98.5%, for abnormal resampled RMS signals (and for normal resampled RMS signals under torque loads of 1 hp, 2 hp, and 3 hp), accuracy in signal modeling decreases. So, the signal estimation technique is suggested in this work. The proposed algorithm is used to estimate the resampled RMS signal using the following steps. First, the resampled RMS signals are estimated using the AMRXU-PI observer, and its mathematical state-space equations are extracted. In the second step, to improve the accuracy of the AMRXU-PI observer, combining the integral term and the AMRXU-PI observer designs the AMRXU-PMI observer. Third, to increase the effect of robustness in the AMRXU-PMI observation technique, the VS-Lyapunov algorithm is used alongside the AMRXU-PMI observer, making the AMRXU-VSPMI observer. In the last step, to improve accuracy and flexibility in uncertain situations, an intelligent technique based on the SNFS is used along with AMRXU-VSPMI, which builds the proposed method: the AMRXU-AVSPMI observer. Thus, the state-space equations for the AMRXU-PI observer are as follows:  
\[ \begin{array}{ll}X_{AMRXU-PI}(k+1)&=[\ell_{X}X_{AMRXU-PI}(k)+\ell_{\epsilon} \epsilon_{AMRXU-PI}(k)+\ell_{MBR}h_{MBR}]\\ &+\ell_{\epsilon}e_{AMRXU-PI}(k)+\epsilon_{AMRXU-PI}+\ell_{p}(Y_{AMRXU-PI}(k )-Y_{AMRXU}(k)),\end{array}  \tag{20}\]  
\[Y_{AMRXU-PI}(k)=\ell_{Y}{}^{T}X_{AMRXU-PI}(k), \tag{21}\]  
\[e_{AMRXU-PI}(k)=Y_{u,rms}(k)-Y_{AMRXU-PI}(k-1), \tag{22}\]  
\[\epsilon_{AMRXU-PI}(k+1)=\epsilon_{AMRXU-PI}(k)+\ell_{i(1)}(\epsilon_{AMRXU -PI}(k)-\epsilon_{AMRXU}(k)). \tag{23}\]  
Here, \(X_{AMRXU-PI}(k)\), \(Y_{AMRXU-PI}(k),e_{AMRXU-PI}(k)\), \(\epsilon_{AMRXU-PI}\), \(Y_{u,rms}(k)\), and \(\left(\ell_{p},\ell_{i(1)}\right)\), 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-PI observation technique), the unknown resampled RMS signals, and the state-space tuning coefficients for proportional and integral terms, also based on the AMRXU-PI observation algorithm. Based on Equation (20), the proportional term is used to increase the accuracy in state estimation of the RMS signal modeling.