## 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)\mathbf{u}_{c}(k)e_{m}(k)\phi_{rg}^{T}[z( k)]\\ w_{\lg}(k+1)-w_{\lg}(k)=-\Omega_{2}(k)\mathbf{u}_{c}(k)e_{m}(k)\phi_{lg}^{T}[z( k)]\end{array}  \tag{44}\]  
_then the uniform stability of the closed loop system is assured and bounded provided that the identification error \(e_{m}(k)\) is within the range:_  
\[\parallel e_{m}(k)\parallel^{2}\geq\frac{2\psi(k)\xi}{\Phi_{1}(k)+\Phi_{2}(k)} \tag{45}\]  
_and the control error satisfies:_  
\[\parallel\mathbf{e}_{c}(k)\parallel_{LI}^{2}\leq\sigma^{2}\left\|Z\right\| \left(1+\frac{(\beta_{1}+\beta_{2})G}{\sigma}\right) \tag{46}\]  
_with the gain \(\sigma\) of the discrete-time sliding mode controller Equation (36) establishing_  
\[\sigma\geq\frac{G}{\left\|K_{G}\right\|}(\beta_{1}+\beta_{2}) \tag{47}\]  
Proof of Theorem 3.1.: The Lyapunov candidate function \(L(k)\) is selected as:  
\[L(k)=\frac{1}{2}\parallel\tilde{\omega}_{rf}(k)\parallel^{2}+\frac{1}{2} \parallel\tilde{\omega}_{lf}(k)\parallel^{2}+\frac{1}{2}\parallel\tilde{ \omega}_{rg}(k)\parallel^{2}+\frac{1}{2}\parallel\tilde{\omega}_{\lg}(k) \parallel^{2}+\frac{1}{\sigma^{2}}\mathbf{e}_{c}^{T}(k)Z\mathbf{e}_{c}(k) \tag{48}\]  
Now for simplicity, \(L(k)=L_{1}(k)+L_{2}(k)\); therefore:  
\[ \begin{array}{l}L_{1}(k)=\frac{1}{2}\parallel\tilde{\omega}_{rf}(k) \parallel^{2}+\frac{1}{2}\parallel\tilde{\omega}_{lf}(k)\parallel^{2}+\frac{1 }{2}\parallel\tilde{\omega}_{rg}(k)\parallel^{2}+\frac{1}{2}\parallel\tilde{ \omega}_{\lg}(k)\parallel^{2}\\ L_{1}(k)=\frac{1}{2}tr\left[\tilde{\omega}_{rf}^{T}(k)\tilde{\omega}_{rf}(k) \right]+\frac{1}{2}tr\left[\tilde{\omega}_{lf}^{T}(k)\tilde{\omega}_{lf}(k) \right]\\ \hskip 14.226378 pt+\frac{1}{2}tr\left[\tilde{\omega}_{rg}^{T}(k)\tilde{\omega}_{ rg}(k)\right]+\frac{1}{2}tr\left[\tilde{\omega}_{\lg}^{T}(k)\tilde{ \omega}_{\lg}(k)\right]\end{array}  \tag{49}\]Now \(\Delta L_{1}(k)=L_{1}(k+1)-L_{1}(k)\),  
\[ \begin{array}{c}\Delta L_{1}(k)=\frac{1}{2}[\parallel\tilde{\omega}_{rf}(k+1) \parallel^{2}-\parallel\tilde{\omega}_{rf}(k)\parallel^{2}]+\frac{1}{2}[ \parallel\tilde{\omega}_{lf}(k+1)\parallel^{2}-\parallel\tilde{\omega}_{lf}( k)\parallel^{2}]\\ +\frac{1}{2}[\parallel\tilde{\omega}_{fg}(k+1)\parallel^{2}-\parallel\tilde{ \omega}_{rg}(k)\parallel^{2}]+\frac{1}{2}[\parallel\tilde{\omega}_{\lg}(k+1) \parallel^{2}-\parallel\tilde{\omega}_{\lg}(k)\parallel^{2}]\end{array}  \tag{50}\]  
From the updated Equation (27),  
\[ \begin{array}{c}\Delta L_{1}(k)=\frac{1}{2}[\parallel\omega_{rf}(k)-\Omega_ {1}(k)e_{i}(k)\phi_{rf}^{T}[z(k)]\parallel^{2}-\parallel\tilde{\omega}_{rf}( k)\parallel^{2}]\\ +\frac{1}{2}[\parallel\omega_{lf}(k)-\Omega_{2}(k)e_{i}(k)\phi_{lf}^{T}[z(k) ]\parallel^{2}-\parallel\tilde{\omega}_{lf}(k)\parallel^{2}]\\ +\frac{1}{2}[\parallel\omega_{rg}(k)-\Omega_{1}(k)u(k)e_{i}(k)\phi_{rg}^{T}[z (k)]\parallel^{2}-\parallel\tilde{\omega}_{rg}(k)\parallel^{2}]\\ +\frac{1}{2}[\parallel\omega_{\lg}(k)-\Omega_{2}(k)u(k)e_{i}(k)\phi_{rg}^{T}[z (k)]\parallel^{2}-\parallel\tilde{\omega}_{\lg}(k)\parallel^{2}]\\ =\frac{1}{2}\Omega_{1}^{2}(k)\parallel e_{m}(k)\parallel^{2}[\parallel\phi_{ rf}^{T}[z(k)]\parallel^{2}-\Omega_{1}(k)\parallel e_{m}(k)\parallel[\parallel \tilde{\omega}_{rf}(k)\phi_{rf}^{T}[z(k)]\parallel\\ +\frac{1}{2}\Omega_{2}^{2}(k)\parallel e_{m}(k)\parallel^{2}[\parallel\phi_{ rf}^{T}[z(k)]\parallel^{2}-\Omega_{2}(k)\parallel e_{m}(k)\parallel[ \parallel\tilde{\omega}_{lf}(k)\phi_{lf}^{T}[z(k)]\parallel\\ +\frac{1}{2}\Omega_{1}^{2}(k)\parallel e_{m}(k)\parallel^{2}[\parallel\phi_{ rf}^{T}[z(k)]\textbf{u}(k)\parallel^{2}-\Omega_{1}(k)\parallel e_{m}(k) \parallel[\parallel\tilde{\omega}_{rg}(k)\phi_{rg}^{T}[z(k)]\textbf{u}_{c}(k) \parallel\\ +\frac{1}{2}\Omega_{2}^{2}(k)\parallel e_{m}(k)\parallel^{2}[\parallel\phi_{ lg}^{T}[z(k)]\textbf{u}(k)\parallel^{2}-\Omega_{2}(k)\parallel e_{m}(k) \parallel[\parallel\tilde{\omega}_{lg}(k)\phi_{lg}^{T}[z(k)]\textbf{u}_{c}(k) \parallel\end{array}