# A Vibration Based Automatic Fault Detection Scheme for Drilling Process Using Type-2 Fuzzy Logic
## 2 Type-2 Fuzzy Logic Modeling Technique
\begin{split}+\frac{1}{2}\omega_{ui}^{*}(k)\alpha_{ui}^{T}(Z_{dis }(k))+(\epsilon_{li}+\epsilon_{ui}),\\ \sum_{i=1}^{p}\omega_{im}^{*}(Z_{dis}(k))=\frac{1}{2}\omega_{lm}^ {*}(k)\alpha_{lm}^{T}(Z_{dis}(k))\\ +\frac{1}{2}\omega_{um}^{*}(k)\alpha_{um}^{T}(Z_{dis}(k))+( \epsilon_{lm}+\epsilon_{um}),\\ \sum_{i=1}^{p}\hat{\omega}_{j}(Z_{dis}(k))=\frac{1}{2}\omega_{lj}^ {*}(k)\alpha_{lj}^{T}(Z_{dis}(k))\\ +\frac{1}{2}\omega_{uj}^{*}(k)\alpha_{uj}^{T}(Z_{dis}(k))+( \epsilon_{lj}+\epsilon_{uj}).\end{split} \]  
The error dynamics are from (35) and (37),  
\[ \begin{split}(\sigma_{1}+\sigma_{2})e_{\text{mod}}(k+1)=\sum_{i=1 }^{p}\hat{\omega}_{i}(Z_{dis}(k))\\ +\sum_{i=1}^{p}\hat{\omega}_{m}(Z_{dis}(k))+\sum_{i=1}^{p}\hat{ \omega}_{j}(Z_{dis}(k))[B_{d}u_{c}(k)]\end{split}  \tag{38}\]\[(\sigma_{1}+\sigma_{2})e_{\text{mod}}(k+1)=\tfrac{1}{2}\bar{\omega}_{ ll}(k)\mathfrak{a}_{li}^{T}(Z_{dis}(k))\] \[+\tfrac{1}{2}\bar{\omega}_{ui}(k)\mathfrak{a}_{ui}^{T}(Z_{dis}(k))\] \[+\tfrac{1}{2}\bar{\omega}_{lm}(k)\mathfrak{a}_{lm}^{T}(Z_{dis}(k))\] \[+\tfrac{1}{2}\bar{\omega}_{lm}(k)\mathfrak{a}_{lm}^{T}(Z_{dis}(k)) \tag{39}\] \[+\tfrac{1}{2}\bar{\omega}_{ij}(k)\mathfrak{a}_{lj}^{T}(Z_{dis}(k) )[B_{d}u_{c}(k)]\] \[+\tfrac{1}{2}\bar{\omega}_{uj}(k)\mathfrak{a}_{uj}^{T}(Z_{dis}(k) )[B_{d}u_{c}(k)]\] \[+\bar{\mathfrak{s}}_{i}+\bar{\varsigma}_{m}+\bar{\varsigma}_{j}[B_ {d}u_{c}(k)]\]  
where \(\bar{\omega}_{li}(k)=\bar{\omega}_{li}(k)-\omega_{li}^{s}(k),\bar{\omega}_{ui} (k)=\bar{\omega}_{ui}(k)-\omega_{ui}^{s}(k),\bar{\omega}_{lm}(k)=\bar{\omega}_ {lm}(k)-\omega_{lm}^{s}(k)\), \(\bar{\omega}_{um}(k)=\bar{\omega}_{um}(k)-\omega_{um}^{s}(k)\), \(\bar{\omega}_{lj}(k)=\bar{\omega}_{lj}(k)-\omega_{lj}^{s}(k),\bar{\omega}_{uj }(k)=\bar{\omega}_{uj}(k)-\omega_{uj}^{s}(k)\), \(\bar{\varsigma}_{i}=\epsilon_{li}+\epsilon_{ui}+R_{i}\,\xi_{m}=\epsilon_{lm}+ \epsilon_{um}+R_{m}\) and \(\xi_{j}=\epsilon_{lj}+\epsilon_{uj}+R_{j}\), \(R_{i},R_{m}\) and \(R_{j}\) are the remainders of the Taylor formula for \(\sum\limits_{i=1}^{p}\bar{\omega}_{i}(Z_{dis}(k))\), \(\sum\limits_{i=1}^{p}\bar{\omega}_{m}(Z_{dis}(k))\) and \(\sum\limits_{i=1}^{p}\bar{\omega}_{j}(Z_{dis}(k))\), respectively. The next theorem provides proof of the stability of the fuzzy modeling. It is for the identification of the nonlinear system using Type 2 fuzzy thus validating that identification error \(e_{\text{mod}}(k)\) is bounded. Theorem 3.1: _If the nonlinear system (26) is compensated using the fuzzy model (35) by utilizing the updating laws (31), then the modeling error \(e_{\text{mod}}(k)\) should be_  
\[\lim_{k\rightarrow\infty}\parallel e_{\text{mod}}(k)\parallel^{2}=\frac{2 \eta(k)\xi}{\pi_{1}(k)+\pi_{2}(k)}\]  
_satisfying the condition \((\sigma_{1}+\sigma_{2})\parallel e_{\text{mod}}(k+1)\parallel>\parallel e_{ \text{mod}}(k)\parallel\)._  
Proof: The candidate \(LP(k)\) for Lyapunov analysis is chosen as follows:  
\[ \begin{array}{l}LP(k)=\tfrac{1}{2}\parallel\bar{\omega}_{li}(k)\parallel^{ 2}+\tfrac{1}{2}\parallel\bar{\omega}_{ui}(k)\parallel^{2}\\ \\ +\tfrac{1}{2}\parallel\bar{\omega}_{lm}(k)\parallel^{2}+\tfrac{1}{2}\parallel \bar{\omega}_{um}(k)\parallel^{2}\\ \\ +\tfrac{1}{2}\parallel\bar{\omega}_{lj}(k)\parallel^{2}+\tfrac{1}{2}\parallel \bar{\omega}_{uj}(k)\parallel^{2}\\ \\ LP(k)=\tfrac{1}{2}tr\big{[}\bar{\omega}_{li}^{T}(k)\bar{\omega}_{li}(k)\big{]} +\tfrac{1}{2}tr\big{[}\bar{\omega}_{ui}^{T}(k)\bar{\omega}_{ui}(k)\big{]}\\ \\ +\tfrac{1}{2}tr\big{[}\bar{\omega}_{lm}^{T}(k)\bar{\omega}_{lm}(k)\big{]}+ \tfrac{1}{2}tr\big{[}\bar{\omega}_{um}^{T}(k)\bar{\omega}_{um}(k)\big{]}\\ \\ +\tfrac{1}{2}tr\big{[}\bar{\omega}_{lj}^{T}(k)\bar{\omega}_{lj}(k)\big{]}+ \tfrac{1}{2}tr\Big{[}\bar{\omega}_{uj}^{T}(k)\bar{\omega}_{uj}(k)\Big{]}\end{array}  \tag{40}\]In addition, the following relation is valid \(\Delta LP(k)=LP(k+1)-LP(k)\), so:  
\[