## 3 Proposed Scheme
### Proposed Robust Smooth Sliding Digital Twin
#### 3.2.2 Proposed Higher-Order Sliding Fuzzy Observer
 \tag{19}\]  
where \(U_{f}(k)\) is the uncertainty estimation using the fuzzy algorithm, and \(C_{\text{f}}\) is the fuzzy coefficient used to reduce the error of the uncertainty estimation and decrease the effect of chattering. The uncertainty estimation using the fuzzy technique is defined using the following definition. \begin{tabular}{l l} \hline \hline \multicolumn{3}{c}{RMS signal estimation using the combination of the proposed autoregressive fuzzy} \\
1: & Gauss-Laguerre RMS signal modeling approach and second-order sliding observer; \\  & Equation (12) \\  & **Detail** \\  & **Calculate** \\
1.1 & \(U_{(aff_{S}-l)-s}(k)\gets C_{\text{u(s1)}}\big{(}S_{i}(k)-\varepsilon_{(aff _{S}-l)-s}(k)\big{)}+C_{\text{u(s2)}}sgn\big{(}S_{i}(k)-\varepsilon_{(aff_{S}- l)-s}(k)\big{)}\,,\) \\  & Equation (13) \\
1.2 & \(\begin{array}{l}\text{Calculate }\varnothing_{(aff_{S}-l)-s}(k+1)\leftarrow\Big{[}C_{ \varnothing}\varnothing_{(aff_{S}-l)-s}(k)+C_{\text{f}}S_{i}(k)+C_{\text{t}} \,\varepsilon_{(aff_{S}-l)-s}(k)\Big{]}+e_{aff_{S}-l}(k)\\ +U_{(aff_{S}-l)-s}(k)+C_{\text{f}}f(k)+C_{\text{u(s)}}sgn|U_{(aff_{S}-l)-s}(k) \,,\) \\
1.3 & \(\begin{array}{l}\text{Calculate }\varepsilon_{(aff_{S}-l)-s}(k)\leftarrow(C_{\text{o}})^{ \text{T}}\chi_{\text{g}}{}^{-1}\varnothing_{(aff_{S}-l)-s}(k)\,,\text{ Equation (\ref{eq:12})}\end{array}\) \\
2: & \(\begin{array}{l}\text{Improve the robustness of RMS signal estimation using the combination of the proposed autoregressive fuzzy Gauss-Laguerre RMS signal modeling approach and high-order}\\ \text{\bf Detail}\\ \\ \text{Equation (\ref{eq:14})}\end{array}\) \\
2.1 & \(\begin{array}{l}\text{Calculate }\lambda(k)\leftarrow C_{\text{o}}\|\big{(}S_{i}(k)- \varepsilon_{(aff_{S}-l)-s}(k)\big{)}\|^{0.5}+C_{\text{p}}sgn\big{(}S_{i}(k)- \varepsilon_{(aff_{S}-l)-s}(k)\big{)}\,,\\ \\ \text{Equation (\ref{eq:15})}\end{array}\) \\
2.2 & \(\begin{array}{l}\text{Calculate }\Gamma(k)\leftarrow-(C_{\text{o}}\|\big{(}S_{i}(k)- \varepsilon_{(aff_{S}-l)-s}(k)\big{)}\,\|^{0.5}+C_{\text{p}}sgn\big{(}S_{i}(k)- \varepsilon_{(aff_{S}-l)-s}(k)\big{)}),\text{ Equation (\ref{eq:15})}\\ \\ \text{Equation (\ref{eq:15})}\end{array}\) \\
2.3 & \(\begin{array}{l}\text{Calculate }\Gamma(k)\leftarrow C_{\text{p}}sgn\big{(}S_{i}(k)- \varepsilon_{(aff_{S}-l)-s}(k)\big{)}\,,\text{ Equation (\ref{eq:15})}\\ \\ \text{Calculate }U_{(aff_{S}-l)-bs}(k)\gets C_{\text{u(s1)}}\big{(}S_{i}(k)- \varepsilon_{(aff_{S}-l)-bs}(k)\big{)}+C_{\text{u(s1)}}\big{(}S_{i}(k)- \varepsilon_{(aff_{S}-l)-bs}(k)\big{)}\,+\\ \\ \text{2.4}\end{array}\) \\
2.4 & \(C_{\text{u(s2)}}sgn\big{(}S_{i}(k)-\varepsilon_{(aff_{S}-l)-bs}(k)\big{)}+C_{ \text{o}}\|\big{(}S_{i}(k)-\varepsilon_{(aff_{S}-l)-s}(k)\big{)}\|^{0.5}+\\ \\ \text{C}_{\text{p}}sgn\big{(}S_{i}(k)-\varepsilon_{(aff_{S}-l)-s}(k)\big{)}-\Gamma( k)\,,\text{ Equation (\ref{eq:16})}\end{array}\) \\
2.5 & \(\begin{array}{l}\text{Calculate }\varnothing_{(aff_{S}-l)-bs}(k+1)\leftarrow\Big{[}C_{ \varnothing}\varnothing_{(aff_{S}-l)-bs}(k)+C_{\text{f}}S_{i}(k)+C_{\text{t}} \,\varepsilon_{(aff_{S}-l)-bs}(k)\Big{]}\\ \\ \text{2.5}\end{array}\) \\
2.5 & \(\begin{array}{l}\text{Calculate }\varepsilon_{(aff_{S}-l)-bs}(k)+C_{\text{f}}f(k)+C_{ \text{u(s)}}sgn\big{|}U_{(aff_{S}-l)-bs}(k)\big{|}^{\frac{2}{3}}\,,\text{ Equation (\ref{eq:16})}\end{array}\) \\
2.6 & \(\begin{array}{l}\text{Calculate }\varepsilon_{(aff_{S}-l)-bs}(k)\leftarrow(C_{\text{o}})^{ \text{T}}\chi_{\text{g}}{}^{-1}\varnothing_{(aff_{S}-l)-bs}(k)\,,\text{ Equation (\ref{eq:16})}\end{array}\) \\ \hline \hline \end{tabular}   
\[U_{f}(k)=\frac{\sum_{z}U_{z}\times\Theta_{z}}{\sum_{z}\Theta_{z}},\Theta_{z}= \sum_{z}e^{\left(-0.5\frac{\sum(\frac{\theta(k)-\mu_{z}}{\sqrt{k}})^{2}}{\frac{ \theta(k)-\mu_{z}}{\sqrt{k}}}\right)} \tag{20}\]  
where \(U_{z}(k)\) is the uncertainty, \(\mu_{z}\) is a membership function used to calculate the fuzzy estimation, and \(\chi_{g}\) is the covariance matrix. Algorithm 2 shows the steps used to design the proposed smooth sliding digital twin to overcome fluctuations arising in unknown conditions.