# Bearing Fault Identification Using Machine Learning and Adaptive Cascade Fault Observer
## Appendix A  
The stability and convergence of the proposed technique is proven in the following part.  
If the fault (unknown condition) estimation in the proposed FOR-ACO is defined by the following equation:  
\[U_{FOR-ACO}= K_{\alpha-new}(\hat{\varphi}(k)+|S_{\partial}|^{\kappa}\hat{ \varphi}(k))\quad,K_{\alpha-new}>0, \tag{1}\]  
\[S_{\partial}=\hat{\varphi}(k)+K_{\alpha-new}\big{(}Y(k)-\hat{Y}(k)\big{)}+ \sigma_{\alpha}\cdot sgn\times\hat{\varphi}(k)+\sigma_{b}\times\hat{\varphi} _{f}(k) \tag{2}\]  
where \(S_{\partial}\) is sliding surface. In the normal condition (\(\varphi(k)=0\)), the convergence reaching time is calculated based on Equation (1).  
\[\begin{array}{c}T_{conv}=\frac{2}{K_{\alpha-new}}\big{(}S_{\ell}+|S_{\ell}| ^{\kappa+1}\big{)},1<\kappa<2\\ S_{\ell}=K\big{(}Y(k)-\hat{Y}_{FOR-ACO}(k)\big{)}\end{array} \tag{3}\]  
Based on [51], in the first step, we defined the convergence time in the normal condition, Equation (1). Based on Equation (1), the residual signal is converged to zero in a finite time. In the abnormal condition, the compensate variable is defined by [52]  
\[\dot{U}_{FOR-ACO}=\varphi(k)-(K_{\alpha-new}(\hat{\varphi}(k)+|S_{\partial}| ^{\kappa}\hat{\varphi}(k)))\quad,S_{\partial}(0)=S_{\partial 0}, \tag{4}\]Based on [52; 53], to have stability and finite time convergence, the coefficient is bounded as follows:  
\[K_{\alpha-new}>2\times(\varphi(k))-\hat{\varphi}(k)) \tag{10}\]  
Based on the Lyapunov theorem, the Lyapunov of the proposed observer is defined by the following equation [51]:  
\[V_{FOR-ACO}(x)=2K_{\alpha-new}|S_{\beta}|+\frac{1}{2}\phi(k)^{2}+\frac{1}{2}(K_ {\alpha-new}|S|^{k}-\phi(k))^{2} \tag{11}\]  
The derivative of the Lyapunov function is defined by Equation (10) [51].  
\[\begin{array}{c}\dot{V}_{FOR-ACO}(x)=\frac{1}{|S_{\beta}|^{c}}\Big{[}\ S_{ \beta}{}^{x}\quad\phi(k)\ \Big{]}\frac{K_{\alpha-new}}{2}\left[\begin{array}{cc}K_{\alpha-new}{}^{2} &-K_{\alpha-new}\\ -K_{\alpha-new}&1\end{array}\right]\ \left[\begin{array}{c}S_{\beta}{}^{x}\\ \phi(k)\end{array}\right]\\ +\frac{\phi_{PD}(k+1)-\phi(k)}{|S_{\beta}|^{k}}\Big{[}\begin{array}{cc}K_{ \alpha-new}{}^{2}&-K_{\alpha-new}\\ \frac{-K_{\alpha-new}}{2}&\Big{[}\begin{array}{c}S_{\beta}{}^{x}\\ \phi(k)\end{array}\end{array}\right]\end{array} \tag{12}\]  
The band of the fault estimation is defined by the following assumption:  
\[\left|\varphi(k)-\phi(k)\right|\leq\omega|S_{\beta}|^{k} \tag{13}\]  
Here, \(\omega\) is a positive constant. Based on Equations (12) and (13), and [51],  
\[\begin{array}{c}\dot{V}_{FOR-ACO}(x)=\frac{-1}{|S_{\beta}|^{c}}\Big{[}\ S_{ \beta}{}^{x}\quad\phi(k)\ \Big{]}\frac{K_{\alpha-new}}{2}\times\\ \left[\begin{array}{cc}K_{\alpha-new}{}^{2}-\left(\frac{1}{K_{\alpha-new}}+ K_{\alpha-new}\right)\left(\varphi(k)-\phi(k)\right)&-K_{\alpha-new}\\ -(K_{\alpha-new}+2(\varphi(k)-\hat{\varphi}(k))&1\end{array}\right]\times \left[\begin{array}{c}S_{\beta}{}^{x}\\ \hat{\varphi}(k)\end{array}\right]\\ \end{array} \tag{14}\]  
So, if \(\left[\begin{array}{cc}K_{\alpha-new}{}^{2}-\left(\frac{1}{K_{\alpha-new}}+ K_{\alpha-new}\right)\left(\varphi(k)-\phi(k)\right)&-K_{\alpha-new}\\ -(K_{\alpha-new}+2(\varphi(k)-\hat{\varphi}(k))&1\end{array}\right]>0\), \(\dot{V}_{FOR-ACO}(x)<0\). Based on [51; 52; 53], when \(\dot{V}_{FOR-ACO}(x)<0\), the residual signals converge to zero in a finite time.