## 3 Proposed Scheme
### Proposed Robust Smooth Sliding Digital Twin
#### 3.2.2 Proposed Higher-Order Sliding Fuzzy Observer
In the presence of uncertain conditions, the estimation error increases. The twisted algorithm is used to reduce the error of estimation in uncertain conditions. Based on the high-order technique, the twisted algorithm is introduced as follows. \[\left\{ \begin{array}{l}\Gamma(k)=-(C_{\alpha}\|\Big{(}\text{S}_{i}(k)- \varepsilon_{(afg-l)-s}(k)\Big{)}\|^{0.5}+C_{\beta}sgn(\text{S}_{i}(k)- \varepsilon_{(afg-l)-s}(k)))\\ \Gamma(k)=C_{\beta}sgn\Big{(}\text{S}_{i}(k)-\varepsilon_{(afg-l)-s}(k)\Big{)} \end{array} \right. \tag{15}\]  
where \(\Gamma(k)\) is the twisting function. Thus, the combination of the proposed autoregressive fuzzy Gauss-Laguerre RMS signal modeling approach and high-order sliding observer is presented as follows. \[\left\{ \begin{array}{l}\varnothing_{(afg-l)-hs}(k+1)=\Big{[}C_{\varnothing \varnothing_{(afg-l)-hs}(k)+C_{i}S_{i}(k)+C_{\epsilon}\varepsilon_{(afg-l)-hs} (k)\Big{]}\\ +e_{afg-l}(k)+U_{(afg-l)-hs}(k)+C_{\text{f}}f(k)+C_{\text{u(s)}}sgn\Big{|}U_{( afg-l)-hs}(k)\Big{|}^{\frac{2}{3}}\\ \\ \varepsilon_{(afg-l)-hs}(k)=(C_{o})^{T}\chi_{g}{}^{-1}\varnothing_{(afg-l)-hs} (k)\end{array} \right. \tag{16}\]  
where \(\varnothing_{(afg-l)-hs}(k)\) is the state of RMS feature estimation using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm and high-order sliding observer, \(U_{(afg-l)-hs}(k)\) is the uncertainty estimation using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm and high-order sliding observer, and \(\varepsilon_{(afg-l)-hs}(k)\) is the output function of RMS feature signal estimation using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm and high-order sliding observer. Furthermore, the uncertainties using the combination of the autoregressive fuzzy Gauss-Laguerre algorithm, high-order sliding observer, and twisting algorithm are described using the following explanation. \[ \begin{array}{l}U_{(afg-l)-hs}(k)= C_{\text{u(s1)}}\Big{(}\text{S}_{i}(k)- \varepsilon_{(afg-l)-hs}(k)\Big{)}+C_{\text{u(s2)}}sgn\Big{(}\text{S}_{i}(k)- \varepsilon_{(afg-l)-hs}(k)\Big{)}\\ +C_{\alpha}\|\Big{(}\text{S}_{i}(k)-\varepsilon_{(afg-l)-s}(k)\Big{)}\|^{0.5}+ C_{\beta}sgn\Big{(}\text{S}_{i}(k)-\varepsilon_{(afg-l)-s}(k)\Big{)}- \Gamma(k)\end{array}  \tag{17}\]  
To condense the consequence of high-frequency oscillation in the combined autoregressive fuzzy Gauss-Laguerre algorithm and high-order sliding observer, a fuzzy algorithm is recommended. Combining the autoregressive fuzzy Gauss-Laguerre algorithm and high-order sliding fuzzy observer (in the proposed smooth sliding digital twin) can be used to overcome the fluctuations arising in uncertain/unknown conditions. Consequently, the proposed smooth digital twin is obtainable, as described by the following definition. \[\left\{ \begin{array}{l}\varnothing_{SSDT}(k+1)=[C_{\varnothing}\varnothing_{SSDT }(k)+C_{\text{f}}S_{i}(k)+C_{\text{t}}\,\varepsilon_{SSDT}(k)]\\ +\varepsilon_{aff-l}(k)+U_{SSDT}(k)+C_{\text{f}}f(k)+C_{\text{u(s)}}sgn|U_{SSDT }(k)|^{\frac{2}{3}}\\ \\ \varepsilon_{SSDT}(k)=(C_{\text{o}})^{\text{T}}\chi_{\text{g}}{}^{-1} \varnothing_{SSDT}(k)\end{array} \right. \tag{18}\]  
where \(\varnothing_{SSDT}(k)\) is the state of RMS feature estimation using the proposed smooth sliding digital twin, \(U_{SSDT}(k)\) is the uncertainty estimation using the proposed smooth sliding digital twin, and \(\varepsilon_{SSDT}(k)\) is the output function of RMS feature signal estimation using the proposed smooth sliding digital twin. Additionally, the uncertainties using the proposed smooth sliding digital twin are described using the following explanation. \[ \begin{array}{l}U_{SSDT}(k)=\\ C_{\text{u(s1)}}(S_{i}(k)-\varepsilon_{SSDT}(k))+C_{\text{u(s2)}}sgn(S_{i}(k)- \varepsilon_{SSDT}(k))+C_{\text{u(s2)}}sgn\big{(}S_{i}(k)-\varepsilon_{SSDT} (k)\big{)}\|^{0.5}\\ +C_{\text{p}}sgn(S_{i}(k)-\varepsilon_{SSDT}(k))+C_{\text{f}}U_{f}(k)-\Gamma(k )\end{array}