## 3 Fault Diagnosis: Machine Learning-based Adaptive Neuro High-Order Variable Structure Observer
### Adaptive Neuro High-Order Variable Structure Observer
\right]\) are the triangular and linguistic variables, which are defined as negative high (NeH), negative medium (NeM), negative low (NeL), zero (Ze), positive low (PoL), positive medium (PoM) and positive high (PoH). The fuzzy membership functions for tuning the fault estimation coefficients of (e) in the interval \(\left[ \begin{array}{cc}-1.5,&1.5\end{array} \right]\) are the triangular and linguistic variables, which are defined as NEH, NEM, NEL, ZE, POL, POM and POH. In addition, the fuzzy linguistic variables for (\(\Upsilon\)) in the interval \(\left[ \begin{array}{cc}0.15,&3\end{array} \right]\) are the Gaussian and fuzzy sets, which are defined as positive small (PoS), positive medium (PoM), and positive big (PoB). An example of the fuzzy If-Then rule is defined by the following equation. Table 1 illustrates the fuzzy rule table used for tuning the coefficients to design the ANHVSO. \[If\ e\ is\ NeH\ and\ e\ is\ NeH\ Then\ \Upsilon\ is\ PoB. \tag{19}\]  
Consequently, the following equation is used to represent the ANHVSO in the robot manipulator to improve the estimation accuracy of the NHVSO:  
\[\left\{ \begin{array}{c}\hat{X}_{1-ANHVSO}=\hat{X}_{2}+\mu_{1}sgn\left(X_{1} -\hat{X}_{1}\right)^{\frac{2}{3}}sgn\left(X_{1}-\hat{X}_{1}\right),\\ \hat{X}_{2-ANHVSO}=\hat{\alpha}^{-1}\left(\hat{X}_{1}\right)\hat{\tau}_{LP,NN +}\hat{\alpha}^{-1}\left(\hat{X}_{1}\right)\hat{\beta}\left(\begin{array}{cc }\hat{X}_{1}&\hat{X}_{2}\end{array}\right)+\hat{\gamma}\left(\hat{X}_{1} \right)+\hat{\alpha}^{-1}\hat{\Theta}_{ANHVSO}+\\ \mu_{2}sgn\left(\hat{X}_{1}-\hat{X}_{2}\right)+\mu_{6}\|X_{1}-\hat{X}_{1}\|^{ \frac{1}{2}}sgn\left(\hat{X}_{1}-\hat{X}_{2}\right)-\Delta,\\ \Delta=-\mu_{5}sgn\left(X_{1}-\hat{X}_{1}\right),\\ \hat{Y}_{ANHVSO}=\left(\epsilon^{T}\right)\hat{X}_{1}\end{array} \right., \tag{20}\]  
The fault is estimated based on the ANHVSO and denoted by the following:  
\[\hat{\Theta}_{ANHVSO}=\mu_{3-New}sgn\left(X_{1}-\hat{X}_{1}\right)+\mu_{7-New }\|X_{1}-\hat{X}_{1}\|^{\frac{1}{2}}sgn\left(X_{1}-\hat{X}_{1}\right). \tag{21}\]  
Here, \(\left(\hat{X}_{1-ANHVSO},\hat{X}_{2-ANHVSO}\right),\hat{Y}_{ANHVSO}\) and \(\hat{\Theta}_{ANHVSO}\) are the observation state estimation, the estimated output, and the fault estimation based on the ANHVSO technique, respectively. When the accuracy of the signal estimation increases, the estimated output \(\left(\hat{Y}_{ANHVSO}\right)\) converges to the measured output \(Y\). This means that the difference between these two signals converges to zero. Thus, the residual signal can be calculated as follows:  
\[r=Y-\hat{Y}_{ANHVSO}, \tag{22}\]  
where \(r\) is the residual signal. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & **Change of Error \(\hat{e}\)** & & \\ \cline{3-8}  & & **NeH** & **NeH** & **NeM** & **NeL** & **Ze** & **PoL** & **PoM** & **PoH** \\
**error** & **NeM** & PoB & PoB & PoB & PoB & PoB & PoM & PoS \\
**(\(\epsilon\))** & **NeL** & PoB & PoB & PoB & PoM & PoM & PoS \\  & **Ze** & PoB & PoB & PoM & PoM & PoM & PoS & PoS \\  & **PoL** & PoB & PoM & PoM & PoM & PoM & PoS & PoS \\  & **PoM** & PoM & PoM & PoM & PoM & PoS & PoS \\  & **PoH** & PoM & PoM & PoM & PoS & PoS & PoS \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The fuzzy rule table for tuning the coefficients in the adaptive neural high order variable structure observer (ANVSO).