## 3 Discrete Time Sliding Mode Control with Type-2 Fuzzy Compensation
\begin{array}{l}w_{rf}(k+1)-w_{rf}(k)=-\Omega_{1}(k)e_{m}(k)\phi_{rf}^{T}[ z(k)]\\ w_{lf}(k+1)-w_{lf}(k)=-\Omega_{2}(k)e_{m}(k)\phi_{lf}^{T}[z(k)]\\ w_{rg}(k+1)-w_{rg}(k)=-\Omega_{1}(k)\textbf{u}_{c}(k)e_{m}(k)\phi_{rg}^{T}[z(k)] \\ w_{\lg}(k+1)-w_{\lg}(k)=-\Omega_{2}(k)\textbf{u}_{c}(k)e_{m}(k)\phi_{\lg}^{T} [z(k)]\end{array}  \tag{27}\]  
Again, \(\Omega_{1}\left(k\right)\) and \(\Omega_{2}\left(k\right)\) satisfy the following:  
\[ \begin{array}{l}\Omega_{1}\left(k\right)=\left\{\begin{array}{cl}\frac{ \Omega_{1}\left(k\right)}{1+\Phi_{1}(k)}&\text{if }\parallel e_{m}(k+1)\parallel>\frac{1}{\beta_{1}} \parallel e_{m}(k)\parallel\\ 0&\text{if }\parallel e_{m}(k+1)\parallel<\frac{1}{\beta_{1}} \parallel e_{m}(k)\parallel\\ \end{array}\right.\\ \Omega_{2}\left(k\right)=\left\{\begin{array}{cl}\frac{\Omega_{2}\left(k \right)}{1+\Phi_{2}(k)}&\text{if }\parallel e_{m}(k+1)\parallel>\frac{1}{\beta_{2}} \parallel e_{m}(k)\parallel\\ 0&\text{if }\parallel e_{m}(k+1)\parallel<\frac{1}{\beta_{2}} \parallel e_{m}(k)\parallel\end{array}\right.\end{array}  \tag{28}\]  
where \(0<\Omega_{1}(k)\leq 1\) and \(0<\Omega_{2}(k)\leq 1\). Moreover, the dead-zone parameters are illustrated by \(\beta_{1}\) and \(\beta_{2}\). Again,  
\[ \begin{array}{l}\Phi_{1}(k)=\parallel\phi_{rf}^{T}[z(k)]\parallel^{2}+ \parallel\phi_{rg}^{T}[z(k)]\textbf{u}_{c}(k)\parallel^{2}\\ \Phi_{2}(k)=\parallel\phi_{lf}^{T}[z(k)]\parallel^{2}+\parallel\phi_{lg}^{T}[ z(k)]\textbf{u}_{c}(k)\parallel^{2}\end{array}  \tag{29}\]  
Now the modeling error \(e_{m}(k)\) is represented as:  
\[e_{m}(k)=\vartheta(k)-v(k) \tag{30}\]  
where the state of the fuzzy model is represented by \(\vartheta(k)\); therefore:  
\[(\beta_{1}+\beta_{2})\vartheta(k+1)=\hat{j}\left[z(k)\right]+\hat{h}\left[z(k) \right]\textbf{u}_{c}(k) \tag{31}\]  
where \(\beta_{1}\) and \(\beta_{2}\) are positive constant and \(\beta_{1},\beta_{2}>1\), which is a design parameter. For stability analysis, Equation (31) is illustrated as:  
\[ \begin{array}{l}(\beta_{1}+\beta_{2})v(k+1)=\frac{1}{2}\omega_{rf}^{*}(k) \phi_{rf}^{T}[z(k)]+\frac{1}{2}\omega_{lf}^{*}(k)\phi_{lf}^{T}[z(k)]\\ +\frac{1}{2}\omega_{rg}^{*}(k)\phi_{rg}^{T}[z(k)]\textbf{u}_{c}(k)+\frac{1}{2} \omega_{lg}^{*}(k)\phi_{lg}^{T}[z(k)]\textbf{u}_{c}(k)+(\epsilon_{rf}+ \epsilon_{lf})+(\epsilon_{rg}+\epsilon_{lg})u(k)\end{array}  \tag{32}\]  
where the unknown optimal weights are \(\omega_{rf}^{*}(k),\omega_{lf}^{*}(k),\omega_{rg}^{*}(k)\) and \(\omega_{lg}^{*}(k)\). Moreover, \(\epsilon_{rf},\epsilon_{lf},\epsilon_{rg}\) and \(\epsilon_{lf}\) are approximation errors which satisfy the relations \(f=\frac{1}{2}\omega_{rf}^{*}(k)\phi_{rf}^{T}[z(k)]+\frac{1}{2}\omega_{lf}^{*}( k)\phi_{lf}^{T}[z(k)]+(\epsilon_{rf}+\epsilon_{lf})\), \(g=\frac{1}{2}\omega_{rg}^{*}(k)\phi_{rg}^{T}[z(k)]+\frac{1}{2}\omega_{lg}^{*}( k)\phi_{lg}^{T}[z(k)]+(\epsilon_{rg}+\epsilon_{lf})\). Using Equations (31) and (32), the error dynamics can be stated as:  
\[ \begin{array}{l}(\beta_{1}+\beta_{2})e_{m}(k+1)=\frac{1}{2}\bar{w}_{rf}(k) \phi_{rf}^{T}[z(k)]+\frac{1}{2}\bar{w}_{lf}(k)\phi_{lf}^{T}[z(k)]\\ +\frac{1}{2}\bar{w}_{rg}(k)\phi_{rg}^{T}[z(k)]\textbf{u}_{c}(k)+\frac{1}{2}\bar{ w}_{lg}(k)\phi_{lg}^{T}[z(k)]\textbf{u}_{c}(k)+\xi_{f}+\xi_{g}\textbf{u}_{c}(k) \end{array}  \tag{33}\]satisfying the relations  
\[ \begin{array}{l}\tilde{\omega}_{rf}(k)=\omega_{rf}(k)-\omega_{rf}^{*}(k)\\ \tilde{\omega}_{lf}(k)=\omega_{lf}(k)-\omega_{lf}^{*}(k)\\ \tilde{\omega}_{rg}(k)=\omega_{rg}(k)-\omega_{rg}^{*}(k)\\ \tilde{\omega}_{lg}(k)=\omega_{lg}(k)-\omega_{lg}^{*}(k)\end{array}  \tag{34}\]  
where \(\xi_{f}=\epsilon_{rf}+\epsilon_{lf}+r_{f}\) and \(\xi_{g}=\epsilon_{rg}+\epsilon_{lg}+r_{g}\). Moreover, the remainders of the Taylor formula for \(j\) and \(\hat{h}\) are illustrated by \(r_{f}\) and \(r_{g}\), respectively. In case of active vibration control, it considered that \(v^{d}(k)=0\).