## 3 Proposed Scheme
### Proposed Robust Smooth Sliding Digital Twin
#### 3.2.1 Proposed Autoregressive Fuzzy Gauss-Laguerre Feature Modeling  
Based on Figure 4, to design an indirect robust smooth sliding digital twin, feature modeling is the first step. Thus, the proposed autoregressive fuzzy Gauss-Laguerre algorithm is designed to model the known (normal feature signal when the speed of the motor is 300 RPM) RMS feature signal. The proposed autoregressive fuzzy Gauss-Laguerre algorithm is designed using the combination of the following techniques: (a) an autoregressive algorithm, (b) Gaussian methods to improve the power of nonlinearity modeling, (c) a Laguerre filter to overcome the uncertain conditions, and (d) a fuzzy algorithm to modify the power of nonlinearity signal modeling. The autoregressive algorithm in introduced using the following definition. \[\left\{ \begin{array}{l}\varnothing_{a}(k+1)=[C_{\varnothing}\mathcal{O}_{a}(k )+C_{i}S_{i}(k)]+e_{a}(k)+U(k)\\ \\ \varepsilon_{a}(k)=(C_{o})^{T}\mathcal{O}_{a}(k)\end{array} \right. \tag{2}\]  
where \(\varnothing_{a}(k)\) is the state of RMS feature modeling using the autoregressive algorithm, \(S_{i}(k)\) is the RMS feature from the original AE signals of the bearing, \(e_{a}(k)\) is the autoregressive error for modeling the RMS feature signal, \(U(k)\) is the unknown/uncertain state for RMS feature modeling, \(\varepsilon_{a}(k)\) is the output function of RMS feature signal modeling using the autoregressive technique, and \((C_{\varnothing},C_{i},C_{o})\) are coefficients (the state of RMS feature modeling, RMS feature from the original AE signals of bearing, and output function of RMS feature signal modeling) of signal modeling. Furthermore, the error of autoregressive RMS feature signal modeling is calculated from the following equation. \[e_{a}(k)=\varepsilon_{a}(k)-\varepsilon_{a}(k-1) \tag{3}\]  
To reduce the error of RMS feature signal modeling using the autoregressive technique, nonlinear-based signal modeling can be very useful. Thus, the Gaussian method is suggested to enhance the power of nonlinearity modeling. The autoregressive Gaussian algorithm is introduced using the following definition. \[\left\{ \begin{array}{l}\varnothing_{ag}(k+1)=\left[C_{\varnothing}\varnothing_ {ag}(k)+C_{i}S_{i}(k)\right]+e_{ag}(k)+U(k)\\ \\ \varepsilon_{ag}(k)=(C_{o})^{T}\chi_{\text{g}}{}^{-1}\varnothing_{ag}(k)\end{array} \right. \tag{4}\]  
where \(\varnothing_{ag}(k)\) is the state of RMS feature modeling using the autoregressive Gaussian algorithm, \(e_{ag}(k)\) is the error of the autoregressive Gaussian for modeling the RMS feature signal g, \(\varepsilon_{ag}(k)\) is the output function of RMS feature signal modeling using the autoregressive Gaussian technique, and \(\chi_{\text{g}}\) is the covariance matrix for RMS feature modeling using the Gaussian algorithm. Furthermore, the covariance matrix (\(\chi_{\text{g}}\)) for RMS feature modeling in known conditions using the Gaussian algorithm is represented as follows. \[\chi_{\text{g}}=\omega^{2}e^{(-0.5\varnothing_{ag}{}^{T}B^{-1}\varnothing_{ag} )}+V \tag{5}\]  
\[B=diag(N)^{2} \tag{6}\]  
where \(\omega\) is the variance of the RMS feature of the AE signal in known conditions, \(V\) is the noise variance, and \(N\) is the kernel width. Additionally, the error of autoregressive Gaussian RMS feature signal modeling is considered from the following equation. \[e_{ag}(k)=\varepsilon_{ag}(k)-\varepsilon_{ag}(k-1) \tag{7}\]The next challenge of RMS feature (indirect) AE signal modeling is the robustness. To increase the robustness of RMS feature signal modeling and reduce the error of RMS feature signal modeling using the autoregressive Gaussian technique, the Laguerre filter technique is combined with the autoregressive Gaussian algorithm. The autoregressive Gauss-Laguerre procedure is defined using the following description. \[\left\{