## 3 Controller Design
### Controller for the Case without Radial Control
\begin{array}{l}\dot{m}=\gamma_{1}s_{1}^{\mathrm{T}}\overline{\mathbf{A }}_{1},\\ \dot{\mathbf{W}}_{1}=(1-\hat{h}_{a,b}(\mathbf{e}_{p}))\mathbf{\xi}_{1}\mathbf{h}(\mathbf{X}){s_{1}} ^{\mathrm{T}},\\ \dot{\mathbf{\hat{\phi}}}_{1}=(1-\hat{h}_{a,b}(\mathbf{e}_{p}))\mathbf{\xi}_{2}|\mathbf{s}_{1} |,\\ \dot{\delta}_{1}=\hat{h}_{a,b}(\mathbf{e}_{p})\mathbf{\xi}_{3}\mathbf{s}_{1},\end{array} \right. \tag{23}\]  
where \(\gamma_{1}>0\), \(\mathbf{\xi}_{1}\), \(\mathbf{\xi}_{2}\) and \(\mathbf{\xi}_{3}\) are diagonal positive definite matrices. To demonstrate the stability of the closed-loop control system, consider the Lyapunov candidate function. \[V_{1}=(1/2)(\mathbf{s}_{1}^{\mathrm{T}}\mathbf{m}\mathbf{s}_{1}+\gamma_{1}^{-1}\widetilde {m}^{2}+\text{tr}(\widetilde{\mathbf{W}}_{1}^{\mathrm{T}}\mathbf{\xi}_{1}\widetilde{ \mathbf{W}}_{1})+\,\mathbf{\tilde{\phi}}_{1}^{\mathrm{T}}\mathbf{\xi}_{2}^{-1}\mathbf{\tilde{ \phi}}_{1}+\widetilde{\delta}_{1}^{\mathrm{T}}\mathbf{\xi}_{3}^{-1}\widetilde{\delta }_{1}) \tag{24}\]  
where \(\widetilde{m}=\hat{m}-m\), \(\widetilde{\mathbf{W}}_{1}=\hat{\mathbf{W}}_{1}-\mathbf{W}_{1}{\ast}\), \(\mathbf{\tilde{\phi}}_{1}=\mathbf{\hat{\phi}}_{1}-\mathbf{\phi}_{1}\) and \(\widetilde{\delta}_{1}=\mathbf{\hat{\delta}}_{1}-\mathbf{\delta}_{1}\) are the estimation errors of m, \(\mathbf{W}_{1}{\ast}\), \(\mathbf{\phi}_{1}\) and \(\mathbf{\delta}_{1}\), respectively. Taking the time derivative of Equation (24) and substituting Equation (23) into the derivative of Equation (24) yields. \[\dot{V}_{1}=\mathbf{s}_{1}^{\rm T}m\mathbf{s}_{1}+\gamma_{1}^{-1}\tilde{m} \dot{\tilde{m}}+tr(\widetilde{\mathbf{W}}_{1}^{\rm T}\xi_{1}\dot{\mathbf{W}}_{1})+ \mathbf{\tilde{\phi}}_{1}^{\rm T}\xi_{2}^{-1}\dot{\mathbf{\phi}}_{1}+\widetilde{\delta}_ {1}^{\rm T}\xi_{3}^{-1}\dot{\mathbf{\delta}}_{1} \tag{25}\] \[=\mathbf{s}_{1}^{\rm T}m(\overline{A}_{1}+m^{-1}(\mathbf{U}_{1}+\delta_{1} ))+\gamma_{1}^{-1}\tilde{m}\dot{\tilde{m}}+tr(\widetilde{\mathbf{W}}_{1}^{\rm T} \xi_{1}\dot{\mathbf{W}}_{1})+\mathbf{\tilde{\phi}}_{1}^{\rm T}\xi_{2}^{-1}\dot{\mathbf{\phi }}^{1}+\widetilde{\delta}_{1}^{\rm T}\xi_{3}^{-1}\dot{\mathbf{\delta}}_{1}\] \[=-\mathbf{s}_{1}^{\rm T}(\mathbf{K}_{1}\mathbf{s}_{1}+\mathbf{\eta}_{1}\mathsf{sig} ^{\sigma_{1}}(\mathbf{s}_{1}))+\mathbf{s}_{1}^{\rm T}\delta_{1}-\mathbf{s}_{1}^{\rm T} \dot{\mathbf{\delta}}_{1}+tr(\widetilde{\mathbf{W}}_{1}^{\rm T}\xi_{1}\dot{\mathbf{W}}_{1}) +\mathbf{\tilde{\phi}}_{1}^{\rm T}\xi_{2}^{-1}\dot{\mathbf{\phi}}_{1}+\widetilde{\delta }_{1}^{\rm T}\xi_{3}^{-1}\dot{\mathbf{\delta}}_{1}\] \[=-\mathbf{s}_{1}^{\rm T}(\mathbf{K}_{1}\mathbf{s}_{1}+\mathbf{\eta}_{1}\mathsf{sig }^{\sigma_{1}}(\mathbf{s}_{1}))+\mathbf{s}_{1}^{\rm T}(1-\hbar)(-\tilde{\mathbf{W}}_{1}\mathbf{ h}(x)+\varepsilon_{1}-diag(\mathbf{\dot{\phi}}_{1})\mathsf{sgn}(\mathbf{s}_{1}))+(1-\hbar) \Big{(}\mathbf{s}_{1}^{\rm T}\widetilde{\mathbf{W}}_{1}\mathbf{h}(x)+\mathbf{\tilde{\phi}}_{1} |\mathbf{s}_{1}|\Big{)}\] \[=-\mathbf{s}_{1}^{\rm T}(\mathbf{K}_{1}\mathbf{s}_{1}+\mathbf{\eta}_{1}\mathsf{sig }^{\sigma_{1}}(\mathbf{s}_{1}))+(1-\hbar)(\mathbf{s}_{1}^{\rm T}\varepsilon_{1}-\mathbf{ \phi}_{1}|\mathbf{s}_{1}|)\] \[=-(1-\hbar)\sum\limits_{i=1}^{2}|\mathbf{s}_{1i}|(\phi_{1i}-\varepsilon _{1i}\mathsf{sgn}(\mathbf{s}_{1i}))-\mathbf{s}_{1}^{\rm T}\mathbf{K}_{1}\mathbf{s}_{1}-\mathbf{s}_ {1}^{\rm T}\mathbf{\eta}_{1}\mathsf{sig}^{\sigma_{1}}(\mathbf{s}_{1})\] \[\leq-(1-\hbar)\sum\limits_{i=1}^{2}|\mathbf{s}_{1i}|(\phi_{1i}-\varepsilon _{1i}\mathsf{sgn}(\mathbf{s}_{1i}))-K_{11}\|\mathbf{s}_{1}\|^{2}-\eta_{11}\|\mathbf{s}_{1} \|^{\sigma_{1}+1}<0.\]  
Therefore, the underactuated closed-loop system is asymptotically stable.