# A Vibration Based Automatic Fault Detection Scheme for Drilling Process Using Type-2 Fuzzy Logic
## 2 Type-2 Fuzzy Logic Modeling Technique
In addition, \(Z_{dis}(k)\in R^{n}\) is the systems state vector, \(y_{dis}(k)\in R^{q}\) is the output, \(u_{c}(k)\in R^{m}\) is the system input, and \(v(k)\) is the unknown bounded system uncertainty. The concept of firing strength having the \(ith\) rule is of the following interval sets:  
\[W_{i}(k)=[\omega_{l}(k),\omega_{u}(k)],i=1,2,\ldots\ldots,p \tag{24}\]where  
\[\omega_{l}(k)=\tfrac{1}{2}{\prod_{a=1}^{\overline{\
u}}}\mu_{(l)\hat{M}_{a}^{i}}(f _{a}(k))\geq 0\]  
\[\omega_{u}(k)=\tfrac{1}{2}{\prod_{a=1}^{\overline{\
u}}}\mu_{(u)\hat{M}_{a}^{i}}( f_{a}(k))\geq 0 \tag{25}\]  
\[\mu_{(l)\hat{M}_{a}^{i}}(f_{a}(k))\geq\mu_{(u)\hat{M}_{a}^{i}}(f_{a}(k))\geq 0\]  
\[\omega_{l}(k)\geq\omega_{u}(k)\geq 0,\forall i\]  
in which \(\omega_{l}(x(t)),\omega_{u}(x(t),\mu_{(l)\hat{M}_{a}^{i}}(f_{a}(x(t)))\) and \(\mu_{(u)\hat{M}_{a}^{i}}(f_{a}(x(t))\) illustrates the lower grade of membership, upper grade of membership, lower membership function, and upper membership function, respectively. The IT2 T-S fuzzy model [1,53] can be illustrated as follows:  
\[Z_{dis}(k+1)={\sum_{i=1}^{p}}\tilde{\omega}_{k}(Z_{dis}(k))A_{d}Z(k)+{\sum_{ i=1}^{p}}\tilde{\omega}_{m}(Z_{dis}(k))\]  
\[+{\sum_{i=1}^{p}}\tilde{\omega}_{j}(Z_{dis}(k))[f_{t}(y_{dis}(k),\Phi(k))+B_{ d}u_{c}(k)] \tag{26}\]  
\[y_{dis}(k)={\sum_{i=1}^{p}}\tilde{\omega}_{j}(Z_{dis}(k))C_{i}Z_{dis}(k)\]  
where \({\sum_{i=1}^{p}}\tilde{\omega}_{i}(Z_{dis}(k))=A_{d}\) and \({\sum_{i=1}^{p}}\tilde{\omega}_{m}(Z_{dis}(k))=D_{d}v(k).\) Relying on the stated Equation (20), the weighting functions associated with the type-2 fuzzy are:  
\[ \begin{array}{l}{\sum_{i=1}^{p}}\tilde{\omega}_{k}(Z_{dis}(k))A_{d}Z_{dis}( k)=\tfrac{1}{2}\omega_{li}(k){\alpha}_{li}^{\overline{l}}(Z_{dis}(k))\\ \qquad\qquad+\tfrac{1}{2}\omega_{ui}(k){\alpha}_{ui}^{\overline{l}}(Z_{dis}(k ))\geq 0,\forall i\\ {\sum_{i=1}^{p}}\tilde{\omega}_{m}(Z_{dis}(k))=\tfrac{1}{2}\omega_{lm}(k){ \alpha}_{lm}^{\overline{l}}(Z_{dis}(k))\\ {\sum_{i=1}^{p}}\tilde{\omega}_{m}(k){\alpha}_{mm}^{\overline{l}}(Z_{dis}(k)) \geq 0,\forall m\\ {\sum_{i=1}^{p}}\tilde{\omega}_{j}(Z_{dis}(k))=\tfrac{1}{2}\omega_{lj}(k){ \alpha}_{lj}^{\overline{l}}(Z_{dis}(k))\\ \qquad\qquad+\tfrac{1}{2}\omega_{uj}(k){\alpha}_{uj}^{\overline{l}}(Z_{dis}(k ))\geq 0,\forall j\end{array}  \tag{27}\]  
In this stage, it is important to prove that the identification error is bounded; therefore, the fault dynamics are considered to be constant and therefore:  
\[{\sum_{i=1}^{p}}\tilde{\omega}_{j}(Z_{dis}(k))[f_{t}(y_{dis}(k),\Phi(k))]=\Omega \tag{28}\]  
where \(\Omega\) is the positive definite matrix. Thus, using Equations (27) and (28) in Equation (26),  
\[ \begin{array}{l}{Z_{dis}(k+1)={\sum_{i=1}^{p}}\tilde{\omega}_{i}(Z_{dis}(k)) +{\sum_{m=1}^{p}}\tilde{\omega}_{m}(Z_{dis}(k))}\\ \qquad\qquad+{\sum_{j=1}^{p}}\tilde{\omega}_{j}(Z_{dis}(k))[B_{d}u_{c}(k)]+ \Omega\end{array}  \tag{29}\]In addition, the weighting functions satisfy the convex sum property depicted as:  
\[ \begin{split}\sum_{i=1}^{p}\tilde{\omega}_{i}(Z_{dis}(k))=1,\sum_{m= 1}^{p}\tilde{\omega}_{m}(Z_{dis}(k))=1,\\ \sum_{j=1}^{p}\tilde{\omega}_{j}(Z_{dis}(k))=1\end{split}  \tag{30}\]  
\[ \begin{split} 0\leq\alpha_{li}^{T}(Z_{dis}(k))\leq 1,0\leq \alpha_{ui}^{T}(Z_{dis}(k))\leq 1,\forall i\\ 0\leq\alpha_{lm}^{T}(Z_{dis}(k))\leq 1,0\leq\alpha_{um}^{T}(Z_{ dis}(k))\leq 1,\forall i\\ 0\leq\alpha_{lj}^{T}(Z_{dis}(k))\leq 1,0\leq\alpha_{uj}^{T}(Z_{ dis}(k))\leq 1,\forall i\end{split} \]  
in which \(\alpha_{li}(Z_{dis}(k)),\alpha_{ui}(Z_{dis}(k))\), \(\alpha_{lm}(Z_{dis}(k))\), \(\alpha_{um}(Z_{dis}(k))\), \(\alpha_{lj}(Z_{dis}(k))\) and \(\alpha_{uj}(Z_{dis}(k))\) are nonlinear functions, \(\tilde{\omega}_{i}(Z_{dis}(k)),\tilde{\omega}_{m}(Z_{dis}(k))\) and \(\tilde{\omega}_{lj}(Z_{dis}(k))\) are the concerned membership functions. In addition, Equation (27) states the type reduction. In addition, \(\omega_{li}(k),\omega_{ui}(k),\omega_{lm}(k),\omega_{um}(k),\omega_{lj}(k)\) and \(\omega_{uj}(k)\) are the fuzzy weighting functions.