## 3 Controller Design
\[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]1  
where \(f_{i}=[f_{i,1}\ f_{i,2}]\). From [27], we have  
\[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]2 \[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]3  
According to \(\gamma_{1}||\bar{\alpha}_{i,1}||\leq\gamma_{1}\) and \(||f_{i}||\leq f_{i}^{*}\), we have  
\[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]4  
According to [PERSON]'s inequality, then:  
\[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]5 \[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]6 (65) \[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]7 (66) \[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]8Equation (60) can be rewritten as follows:  
\[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]9 \[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]0 \[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]1 \[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]2  
where \(\mu_{1}=\sum_{i=1}^{n}\{(\theta_{i,1}^{2}\gamma_{1}^{2})/2+\gamma_{1}f_{i}^{* }+0.2785\epsilon_{i,1}\}\). Define \(\bar{m}=\max\{\frac{1}{2}m_{i,11}^{2},\frac{1}{2},\frac{1}{2}m_{i,33}^{2}\}\), then  
\[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]3 \[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]4  
Consider the total function  
\[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]5  
Using (12), (68) and (69), the derivative of \(V_{t}\) is given by  
\[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]6 \[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]7 \[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]8  
where \(\Omega_{i,1}=\dot{\beta}_{i,1}-f_{i,1}\), \(\Omega_{i,2}=\dot{\beta}_{i,2}-f_{i,2}\) and \(\Omega_{i,3}=\dot{\beta}_{i,3}+\psi_{i,e}\theta_{i,1}\). According to [PERSON]'s inequality, \(\omega_{i,h}\Omega_{i,h}\leq\frac{1}{2\varrho_{i}}\omega_{i,h}^{2}\Omega_{i,h }^{2}+\frac{\varrho_{i}}{2}\) with \(\varrho_{i}\) being a small positive constant and \(\mu_{2}=\mu_{1}+\frac{3\varrho_{i}}{2}+\sum_{i=1}^{n}\frac{1}{2}(\zeta_{i}^{ *})^{2}\). Consider the sets \(\prod_{i}:\{\sum_{j=1}^{n}q_{j,e}+\psi_{j,e}^{2}+m_{j,11}u_{j,e}^{2}+v_{j,e}^ {2}+m_{j,33}r_{j,e}^{2}+\sum_{h=1}^{3}\omega_{j,h}^{2}+\frac{1}{2}\tilde{X}_{ j}^{T}\tilde{X}_{j}\leq 2\varrho_{\rho}\}\) and \(O_{o}:=\{\eta_{0}^{T}\eta_{0}+\eta_{0}^{T}\eta_{0}\leq\eta_{0}^{*}\}\) where \(\rho_{\rho}\) and \(\eta_{0}^{*}\) are positive constants. Since \(\prod_{i}\times O_{o}\) is compact in \(\mathbb{R}^{8i+6}\), there exist a constant \(s_{i,h}>0\) such that \(|\Omega_{i,h}|\leq s_{i,h}\) on \(\prod_{i}\times O_{o}\).