# A Vibration Based Automatic Fault Detection Scheme for Drilling Process Using Type-2 Fuzzy Logic
## 2 Type-2 Fuzzy Logic Modeling Technique
\begin{array}{l}\Delta LP(k)=\frac{1}{2}\eta^{2}(k)\parallel e_{\text{mod}} (k)\parallel^{2}[\parallel\bar{\alpha}_{li}^{T}(Z_{dis}(k))\parallel^{2}\\ \\ +\parallel a_{lm}^{T}(Z_{dis}(k))\parallel^{2}+\parallel a_{lj}^{T}(Z_{dis}(k))[ B_{d}u_{c}(k)]\parallel^{2}]\\ \\ +\frac{1}{2}\eta^{2}(k)\parallel e_{\text{mod}}(k)\parallel^{2}[ \parallel a_{ui}^{T}(Z_{dis}(k))\parallel^{2}\\ \\ +\parallel a_{um}^{T}(Z_{dis}(k))\parallel^{2}+\parallel a_{uj}^{T}(Z_{dis}(k))[ B_{d}u_{c}(k)]\parallel^{2}]\\ \\ -2\eta(k)\parallel e_{\text{mod}}^{T}(k)\parallel\\ \\ \times[(\sigma_{1}+\sigma_{2})e_{\text{mod}}(k+1)-\xi_{i}-\xi_{m}-\xi_{j}[B_{d}u _{c}(k)]]\\ \\ \Delta LP(k)=\frac{1}{2}\eta^{2}(k)\parallel e_{\text{mod}}(k)\parallel^{2}[ \parallel a_{li}^{T}(Z_{dis}(k))\parallel^{2}\\ \\ +\parallel a_{lm}^{T}(Z_{dis}(k))\parallel^{2}+\parallel a_{lj}^{T}(Z_{dis}(k))[ B_{d}u_{c}(k)]\parallel^{2}]\\ \\ +\frac{1}{2}\eta^{2}(k)\parallel e_{\text{mod}}(k)\parallel^{2}[ \parallel\alpha_{ui}^{T}(Z_{dis}(k))\parallel^{2}\\ \\ +\parallel a_{um}^{T}(Z_{dis}(k))\parallel^{2}+\parallel a_{uj}^{T}(Z_{dis}(k))[ B_{d}u_{c}(k)]\parallel^{2}]\\ \\ -2\eta(k)\parallel e_{\text{mod}}(k)\parallel(\sigma_{1}+\sigma_{2})\parallel e _{\text{mod}}(k+1)\parallel\\ \\ +2\eta(k)\parallel e_{\text{mod}}(k)\parallel[\xi_{i}+\xi_{m}+\xi_{j}[B_{d}u_{c} (k)]]\end{array}  \tag{44}\]  
Using the definition (32), (33) and \((\sigma_{1}+\sigma_{2})\parallel e_{\text{mod}}(k+1)\parallel>\parallel e_{ \text{mod}}(k)\parallel\) and \(\xi(k)=\xi_{i}+\xi_{m}+\xi_{j}[B_{d}u_{c}(k)]\), then:\[ \begin{array}{l}\Delta LP(k)\leq-2\eta(k)\parallel e_{\text{mod}}(k)\parallel^{2} \\ \\ +\frac{1}{2}\eta^{2}(k)\parallel e_{\text{mod}}(k)\parallel^{2}(\pi_{1}(k)+\pi_{ 2}(k))\\ \\ +2\eta(k)\parallel e_{\text{mod}}(k)\xi(k)\parallel\\ \\ \leq-\eta(k)\parallel e_{\text{mod}}(k)\parallel^{2}\\ \\ +\frac{1}{2}\eta^{2}(k)\parallel e_{\text{mod}}(k)\parallel^{2}(\pi_{1}(k)+ \pi_{2}(k))\\ \\ +\eta(k)\parallel\xi(k)\parallel^{2}\\ \\ \leq-\eta(k)\{1-\frac{\eta(k)}{2}(\pi_{1}(k)+\pi_{2}(k)\}\parallel e_{\text{ mod}}(k)\parallel^{2}\\ \\ +\eta(k)\parallel\xi(k)\parallel^{2}\end{array}  \tag{46}\]  
Since the modeling error \(\xi(k)\) is bounded as  
\[\parallel\xi(k)\parallel^{2}\leq\parallel\xi \tag{47}\]  
in order that \(\Delta LP(k)\leq 0\),  
\[ \begin{array}{l}\frac{\eta(k)}{2}(\pi_{1}(k)+\pi_{2}(k)\}\parallel e_{\text {mod}}(k)\parallel^{2}\geq\eta(k)\xi\\ \\ \parallel e_{\text{mod}}(k)\parallel^{2}\geq\frac{2\eta(k)\xi}{\pi_{1}(k)+\pi_{ 2}(k)}\end{array}  \tag{48}\]  
Therefore, it is proved that \(LP(k)\) is bounded. In addition, if \(\Psi_{1}(k),\Psi_{2}(k)=0\), then, from (31), it is evident that the weights do not change and hence they are bounded. In addition, in this stage, it is assumed that the fault dynamics are not changing due to the non occurrence of faults. This theorem validates that the fuzzy modeling approach can be implemented with assured stability. The next stage is the fault detection scheme where an innovative observer will be developed to detect faults in the drilling bit with respect to the vibration measurements.