## 3 Proposed Scheme
### Proposed Robust Smooth Sliding Digital Twin
#### 3.2.2 Proposed Higher-Order Sliding Fuzzy Observer  
After modeling the known RMS feature signal using the proposed autoregressive fuzzy Gauss-Laguerre approach, an observation algorithm based on the proposed higher-order sliding fuzzy technique is designed in the second step. The proposed observer is used to estimate the RMS signal of the bearing in all conditions. To design a proposed observer, a sliding observation algorithm is first suggested. Using a sliding observer is a nonlinear and robust technique to estimate the signals. To improve the robustness of the classical (second-order) sliding observer, a higher-order technique is suggested in this work. Apart from the robustness of the high-order sliding observer, this technique suffers from high-frequency oscillation (chattering phenomenon). To reduce the effect of chattering in the high-order sliding observer, a fuzzy algorithm is suggested. A high-order sliding fuzzy observer can be used to overcome fluctuations arising in uncertain/unknown conditions. First, the combination of the proposed autoregressive fuzzy Gauss-Laguerre RMS signal modeling approach and second-order sliding observer is introduced as follows. \[\left\{ \begin{array}{l}\varnothing_{(afg-l)-s}(k+1)=\left[\mathcal{C}_{ \varnothing}\varnothing_{(afg-l)-s}(k)+C_{i}\mathcal{S}_{i}(k)+C_{\varepsilon} \varepsilon_{(afg-l)-s}(k)\right]\\ \quad+\varepsilon_{afg-l}(k)+U_{(afg-l)-s}(k)+C_{t}f(k)+C_{u(s)}sgnU_{(afg-l)-s}( k)\\ \varepsilon_{(afg-l)-s}(k)=(C_{o})^{T}\chi_{g}{}^{-1}\varnothing_{(afg-l)-s}(k) \end{array} \right. \tag{12}\]  
```
1: RMS feature signal modeling using the autoregressive procedure; Equation (2)
2:Detail
3:Calculate \(e_{a}(k)\leftarrow\varepsilon_{a}(k)-\varepsilon_{a}(k-1)\), Equation (3)
4:Calculate \(\varnothing_{a}(k+1)\leftarrow\left[\mathcal{C}_{\varnothing}\varnothing_{a}(k)+ C_{i}\mathcal{S}_{i}(k)\right]+e_{a}(k)+U(k)\), Equation (2)
5:Calculate \(\varepsilon_{a}(k)\leftarrow(C_{o})^{T}\mathcal{C}_{a}(k)\), Equation (2)
```  
**Algorithm 1**Proposed RMS feature modeling using the autoregressive fuzzy Gauss-Laguerre approach. Currently, \(\varnothing_{(afg-l)-s}(k)\) is the state of RMS feature estimation using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm and second-order sliding observer, \(U_{(afg-l)-s}(k)\) is the uncertainty estimation using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm and second-order sliding observer, \(\varepsilon_{(afg-l)-s}(k)\) is the output function of RMS feature signal estimation using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm and second-order sliding observer, and \(C_{u(s)}\) is the coefficient of the second-order sliding observer for RMS feature estimation using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm and second-order sliding observer. Moreover, the uncertainties using the combination of the autoregressivefuzzy Gauss-Laguerre algorithm and second-order sliding observer are described using the following explanation. \[U_{(afg-l)-s}(k)=C_{\text{u(s1)}}\Big{(}\text{S}_{i}(k)-\varepsilon_{(afg-l)-s}(k) \Big{)}+C_{\text{u(s2)}}sgn\Big{(}\text{S}_{i}(k)-\varepsilon_{(afg-l)-s}(k) \Big{)} \tag{13}\]  
where \(\Big{(}C_{\text{u(s1)}}\) and \(C_{\text{u(s2)}}\Big{)}\) are the coefficients of uncertain estimation using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm and second-order sliding observer. To recover the robustness in the classical sliding observer, the higher-order technique is recommended in this effort. The definition of the high-order sliding observer is introduced using the following explanation. \[\lambda(k)=C_{\alpha}\|\Big{(}\text{S}_{i}(k)-\varepsilon_{(afg-l)-s}(k)\Big{)} \|^{0.5}+C_{\beta}sgn\Big{(}\text{S}_{i}(k)-\varepsilon_{(afg-l)-s}(k)\Big{)} \tag{14}\]  
where \(\lambda(k)\) is the high-order definition for the sliding observer and \((C_{\alpha}\) and \(C_{\beta})\) are the coefficients for RMS signal estimation in the high-order sliding observer.