# A Vibration Based Automatic Fault Detection Scheme for Drilling Process Using Type-2 Fuzzy Logic
## 3 The Technique of Fault Detection
\begin{split}\sum\limits_{i=1}^{p}\tilde{\omega}_{j}(Z_{dis}(k) )C_{d}=0\text{ and }\\ \left(-\sum\limits_{i=1}^{p}\tilde{\omega}_{k}(Z_{dis}(k))A_{d}+ \sum\limits_{i=1}^{p}\tilde{\omega}_{j}(Z_{dis}(k))k_{f}C_{d}\right)=0\end{split}  \tag{59}\]  
The conditions Equations (58) and (59) are utilized to extract the values of observer gains. Now, we propose that \(\tilde{\omega}_{j}(Z_{dis}(k))=\mu\tilde{\omega}_{k}(Z_{dis}(k))\).In addition, \(\hat{\Phi}^{T}\Delta_{i}(k)=\Pi\) and \(A_{d}-\mu k_{f}C_{d}=A_{\tilde{d}}\). Using Equation (55):  
\[ \begin{split} e_{z}(k+1)=\sum\limits_{i=1}^{p}\tilde{\omega}_{k} (Z_{dis}(k))A_{d}e_{z}(k)\\ +\sum\limits_{i=1}^{p}\tilde{\omega}_{m}(Z_{dis}(k))+\sum\limits_ {i=1}^{p}\tilde{\omega}_{j}(Z_{dis}(k))\Pi\\ e_{y}(k)=\sum\limits_{i=1}^{p}\tilde{\omega}_{j}(Z_{dis}(k))C_{d} e_{z}(k)\end{split}  \tag{60}\]  
The following updated laws will be used to verify that the state residual and parameter estimate error are uniformly bounded, which also confirms that the output residual is uniformly bounded:  
\[ \begin{split}\hat{\Phi}_{l}(k+1)=\frac{1}{2}(\hat{\Phi}_{l}(k)+ \hat{\Phi}_{u}(k))\\ -\frac{1}{2}R_{\theta}\sum\limits_{i=1}^{p}\tilde{\omega}_{j}(Z_ {dis}(k))\Delta_{i}(k)e_{y}^{T}(k)B_{\theta}\\ -\frac{1}{2}\gamma\sum\limits_{i=1}^{p}\tilde{\omega}_{j}(Z_{dis }(k))(\[PERSON]_{l}(k)\\ +\hat{\[PERSON]}_{u}(k))(\|I-R_{\theta}\Delta_{i}(k)\Delta_{i}^{T}(k)\|) \\ \hat{\[PERSON]}_{u}(k+1)=\frac{1}{2}(\hat{\[PERSON]}_{u}(k)+\hat{\[PERSON]}_{l}( k))\\ -\frac{1}{2}R_{\theta}\sum\limits_{i=1}^{p}\tilde{\omega}_{j}(Z_ {dis}(k))\Delta_{i}(k)e_{y}^{T}(k)B_{\theta}\\ -\frac{1}{2}\gamma\sum\limits_{i=1}^{p}\tilde{\omega}_{j}(Z_{dis }(k))(\hat{\Phi}_{u}(k)\\ +\Phi_{l}(k))(\|I-R_{\theta}\Delta_{i}(k)\Delta_{i}^{T}(k)\|)\end{split}  \tag{61}\]  
**Theorem 2**. : _The boundary condition of the output residual \(e_{y}(k)\) and the parameter estimation error \(\bar{\Phi}\) will be validated if the conditions stated below are satisfied:_  
\[\parallel e_{z}(k)\parallel\geq\frac{2(\phi_{1}^{2}+\phi_{2}^{2})}{(1-A_{d_{ \max}}^{2}-\frac{1}{4}R_{\beta}^{2}R_{\beta}^{2}\Delta_{\max}^{2}C_{d_{\max}}^{ 2})}\]  
\[\parallel\Phi_{l}(k)+\Phi_{u}(k)\parallel\] \[\geq\sqrt{\frac{(\Psi_{d}++\omega_{1}^{2}+L_{\max}^{2}+2\phi_{3}^ {2})}{\frac{1}{4}+\gamma^{2}\parallel\overset{p}{\underset{i=1}{\sum}}\bar{ \omega}_{j}(Z_{dis}(k))\parallel^{2}\parallel I-R_{\beta}\Delta_{i}(k)\Delta_{ i}^{T}(k)\|^{2}|}}\]  
_when a discrete time observer is used to monitor a nonlinear system that has been updated with new laws. Equations (61) and (62) are utilized for the unknown parameter tuning process linked to ADT_  
Proof. : Consider Lyapunov candidate function as:  
\[ \begin{array}{l}LP(k)=e_{z}^{T}(k)e_{z}(k)\\ \\ \left\{\frac{1}{2}tr\big{[}\bar{\[PERSON]}_{l}^{T}(k)\bar{\[PERSON]}_{l}(k)\big{]}+ \frac{1}{2}tr\big{[}\bar{\[PERSON]}_{u}^{T}(k)\bar{\[PERSON]}_{u}(k)\big{]}\right\}\end{array}  \tag{63}\]  
The first difference of the Equation (63) is given by:  
\[ \begin{array}{l}\Delta LP(k)=\big{[}e_{z}^{T}(k+1)e_{z}(k+1)-e_{z}^{T}(k)e_{ z}(k)\big{]}\\ \\ +\frac{1}{2}tr\big{[}\bar{\[PERSON]}_{l}^{T}(k+1)\bar{\[PERSON]}_{l}(k+1)-\bar{\[PERSON]}_{l }^{T}(k)\bar{\[PERSON]}_{l}(k)\big{]}\\ \\ +\frac{1}{2}tr\big{[}\bar{\[PERSON]}_{u}^{T}(k+1)\bar{\[PERSON]}_{u}(k+1)-\bar{\[PERSON]}_{u} ^{T}(k)\bar{\[PERSON]}_{u}(k)\big{]}\end{array}  \tag{64}\]  
Now, let  
\[ \begin{array}{l}\Delta LP_{1}(k)=\big{[}e_{z}^{T}(k+1)e_{z}(k+1)-e_{z}^{T}( k)e_{z}(k)\big{]}\\ \\ \Delta LP_{2}(k)=\frac{1}{2}tr\big{[}\bar{\[PERSON]}_{l}^{T}(k+1)\bar{\[PERSON]}_{l}(k+ 1)-\bar{\[PERSON]}_{l}^{T}(k)\bar{\[PERSON]}_{l}(k)\big{]}\\ \\ +\frac{1}{2}tr\big{[}\bar{\[PERSON]}_{u}^{T}(k+1)\bar{\[PERSON]}_{u}(k+1)-\bar{\[PERSON]}_{u} ^{T}(k)\bar{\[PERSON]}_{u}(k)\big{]}\end{array}  \tag{65}\]  
Now, \(\Delta LP_{1}(k)\) will be expanded. Using Equation (60):  
\[