## 3 Fault Diagnosis: Machine Learning-based Adaptive Neuro High-Order Variable Structure Observer
### Adaptive Neuro High-Order Variable Structure Observer  
Based on Figure 1 and the robot manipulator model, an adaptive neuro high-order variable structure observer (ANHVSO) can be designed to estimate faults in the presence of uncertainties and external distortion. Based on Equations (5) and (6), the state-space robot manipulator model is represented by the following equation:  
\[\left\{ \begin{array}{c}\dot{X}_{1}=X_{2}=q\\ \dot{X}_{2}=\alpha^{-1}(X_{1})\tau_{LPF}+\alpha^{-1}(X_{1})\beta\Big{(}\begin{array} []{cc}X_{1}&X_{2}\end{array}\Big{)}-\gamma(X_{1})-\alpha^{-1}\Theta\\ Y=X_{1}\end{array} \right.. \tag{7}\]  
Here, \((X_{1},X_{2}),\Big{(}\dot{X}_{1},\dot{X}_{2}\Big{)}\) and \(Y\) are the state estimation, the change of the state estimation, and the measured output state-space signal for the robot manipulator, respectively. The robust classical VSO technique used to estimate the faulty signal is as follows [3,9]. \[\left\{ \begin{array}{c}\dot{\hat{X}}_{1-VSO}=\hat{X}_{2}+\mu_{1}sgn\big{(}X_ {1}-\hat{X}_{1}\big{)}\\ \dot{\hat{X}}_{2-VSO}=\hat{\alpha}^{-1}\Big{(}\hat{X}_{1}\big{)}\hat{\tau}_{LPF }+\hat{\alpha}^{-1}\Big{(}\hat{X}_{1}\big{)}\beta\Big{(}\begin{array}{cc}\hat {X}_{1}&\hat{X}_{2}\end{array}\Big{)}+\hat{\gamma}\Big{(}\hat{X}_{1}\big{)}+ \hat{\alpha}^{-1}\Theta_{VSO}+\mu_{2}sgn\Big{(}\dot{\hat{X}}_{1}-\hat{X}_{2} \Big{)}\\ \hat{Y}_{VSO}=\Big{(}\epsilon^{T}\Big{)}\hat{X}_{1}\end{array} \right. \tag{8}\]  
The VSO for fault (\(\hat{\Theta}\)) estimation is represented by the following equation:  
\[\hat{\Theta}_{VSO}=\mu_{3}sgn\Big{(}X_{1}-\hat{X}_{1}\Big{)}. \tag{9}\]  
Here, \((\mu_{1},\mu_{1},\mu_{3}),\big{(}\hat{X}_{1-VSO},\hat{X}_{2-VSO}\big{)},\hat{Y }_{VSO},\hat{\Theta}_{VSO},\) and \(\epsilon^{T}\) are variable structure coefficients, the observation state estimation based on the VSO, the estimated output based on the VSO, the fault estimation based on the VSO technique, and the coefficient for the output estimation based on the VSO, respectively. To reduce high-frequency oscillations in the VSO, application of the higher-order VSO (HVSO) is recommended. The super-twisting (ST) technique for the HVSO is represented by the following Equation [9]:  
\[\left\{ \begin{array}{c}\varrho=\mu_{4}\|X_{1}-\hat{X}_{1}\|^{\frac{1}{3}}sgn \big{(}X_{1}-\hat{X}_{1}\big{)}-\Delta\\ \dot{\Delta}=-\mu_{5}sgn\Big{(}X_{1}-\hat{X}_{1}\Big{)}\end{array} \right., \tag{10}\]  
Here, \(\varrho,\Delta,\) and \((\mu_{4},\mu_{5})\) are the performance enhancement function of the VSO, a variable in the ST function to reduce the estimation error, and a coefficient, respectively. Thus, the following equation is used to represent the HVSO in a robot manipulator to reduce chattering in the VSO:  
\[\left\{ \begin{array}{c}\dot{\hat{X}}_{1-HVSO}=\hat{X}_{2}+\mu_{1}sgn\big{(} X_{1}-\hat{X}_{1}\big{)}^{\frac{2}{3}}sgn\big{(}X_{1}-\hat{X}_{1}\big{)}\,,\\ \dot{\hat{X}}_{2-HVSO}=\hat{\alpha}^{-1}\Big{(}\hat{X}_{1}\big{)}\hat{\tau}_{LPF }+\hat{\alpha}^{-1}\Big{(}\hat{X}_{1}\big{)}\hat{\beta}\Big{(}\begin{array}{ cc}\hat{X}_{1}&\hat{X}_{2}\end{array}\Big{)}+\hat{\gamma}\Big{(}\hat{X}_{1} \big{)}+\hat{\alpha}^{-1}\hat{\Theta}_{HVSO}+\mu_{2}sgn\Big{(}\dot{\hat{X}}_{1 }-\hat{X}_{2}\Big{)}+\\ \mu_{6}\|X_{1}-\hat{X}_{1}\|^{\frac{1}{3}}sgn\Big{(}\dot{\hat{X}}_{1}-\hat{X}_{ 2}\Big{)}-\Delta,\\ \dot{\Delta}=-\mu_{5}sgn\big{(}X_{1}-\hat{X}_{1}\big{)},\\ \hat{Y}_{HVSO}=\Big{(}\epsilon^{T}\Big{)}\hat{X}_{1}\end{array} \right., \tag{11}\]  
The fault can be estimated and represented using the HVSO as described in the following equation:  
\[\hat{\Theta}_{HVSO}=\mu_{3}sgn\Big{(}X_{1}-\hat{X}_{1}\Big{)}+\mu_{7}\|X_{1}- \hat{X}_{1}\|^{\frac{1}{2}}sgn\Big{(}X_{1}-\hat{X}_{1}\Big{)}. \tag{12}\]Here, \((\mu_{6},\mu_{7}),\big{(}\hat{X}_{1-HVSO},\hat{X}_{2-HVSO}\big{)},\hat{Y}_{HVSO}\) and \(\hat{\Theta}_{HVSO}\) are high-order variable structure coefficients, the observation state estimation based on the HVSO, the estimated output based on the HVSO, and the fault estimation based on the HVSO technique, respectively.