# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## Appendix A  
The stability and convergence of the proposed extended-state ARX-Laguerre PI observer technique is proven in the following part. _Proof_: if the extended-state observer is defined by the following Equation:  
\[\mathrm{U_{hgo}}=\ \ \mathrm{K_{c-new}}\big{(}\hat{\delta}_{\mathrm{PIO}}(\mathrm{k }+1)+|\mathrm{S_{0}}|^{\kappa}\hat{\delta}_{\mathrm{PIO}}(\mathrm{k}+1)\big{)} \quad,\mathrm{K_{c-new}}>0,\] (A-1)  
\[\mathrm{S_{0}}=\hat{\delta}(\mathrm{k})+\mathrm{K}\big{(}\mathrm{Y}(\mathrm{k })-\hat{\mathrm{Y}}(\mathrm{k})\big{)}\] (A-2)  
In the normal condition (\(\delta(k)=0\)), the convergence reaching time is calculated based on (A-3). \[ \begin{array}{l}T_{conv}=\frac{2}{\mathrm{K_{c-new}}}\Big{(}S_{\ell}+|S_{ \ell}|^{\kappa+1}\Big{)},1<\kappa<2\\ S_{\ell}=K\big{(}Y(k)-\hat{\mathrm{Y}}(k)\big{)}\end{array} \] (A-3)  
Based on [29], in the first step, we defined the convergence time in the normal condition, Equation (A-4). Based on Equation (A-4), the residual signal is converged to zero in a finite time. In the abnormal condition, the compensate variable is defined by  
\[\mathrm{\dot{U}_{hgo}}=\delta_{\mathrm{PIO}}(\mathrm{k}+1)-\big{(}\mathrm{K_{c -new}}\big{(}\delta_{\mathrm{PIO}}(\mathrm{k}+1)+|\mathrm{S_{0}}|^{\kappa} \delta_{\mathrm{PIO}}(\mathrm{k}+1)\big{)}\big{)}\quad,\mathrm{S_{0}}(0)= \mathrm{S_{00}},\] (A-4)  
Based on [29], to have stability and finite time convergence, the coefficient is bounded as follows:  
\[\mathrm{K_{c-new}}>2\times(\delta_{\mathrm{PIO}}(\mathrm{k}+1))-\hat{\delta}_ {\mathrm{PIO}}(\mathrm{k}+1))\] (A-5)  
Based on the Lyapunov theorem, the Lyapunov of the proposed observer is defined by the following Equation:  
\[\mathrm{V_{proposed}}(\mathrm{x})=2\mathrm{K_{c-new}}|\mathrm{S_{0}}|+\frac{1}{2} \hat{\delta}_{\mathrm{PIO}}(\mathrm{k}+1)^{2}+\frac{1}{2}(\mathrm{K_{c-new}}| \mathrm{S}|^{\kappa}-\hat{\delta}_{\mathrm{PIO}}(\mathrm{k}+1))^{2}\] (A-6)  
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