## 4 Nonsingular Predefined-Time Attitude Controller Design
\right)\text{, }\mathbf{U}=\mathbf{A}\mathbf{R}\mathbf{J}^{-1}(M_{c}+M_{ air}),\\ \overline{\Xi}=\text{sec}^{2}\bigg{(}\frac{\pi t}{2l_{f}}\bigg{)}\text{, }\ t\in\left[0,\ \overline{t}_{f}\right)\text{, and }\overline{t}_{f}\text{ is the convergence time. The time switching function is  
\[\dot{S} =\big{(}2+\tfrac{2}{3}a\big{)}\text{diag}\Big{(}|e_{1i}|^{1+2a/3} \Big{)}\mathbf{e}_{2}+\mathbf{\Phi}\Big{(}\tfrac{e_{2}}{\Theta}+\big{(}2+\tfrac{2a}{3} \big{)}\text{diag}\Big{(}|e_{1i}|^{1+2a/3}\Big{)}\mathbf{e}_{2}\Big{)} \tag{32}\] \[=\big{(}2+\tfrac{2}{3}a\big{)}\text{diag}\Big{(}|e_{1i}|^{1+2a/3} \Big{)}\mathbf{e}_{2}+\mathbf{\Phi}\left(\begin{array}{c}\tfrac{1}{\Theta}\bigg{(} \mathbf{P}\dot{\mathbf{e}}_{1}+\mathbf{A}\bigg{(} \begin{array}{c}(\mathbf{R}\mathbf{R}^{-1}+\mathbf{ R}\mathbf{J}^{-1}\mathbf{\Omega})\mathbf{R}^{-1})\mathbf{z}_{2}+\mathbf{R}\mathbf{J}^{-1}(\mathbf{M}_{c}+\mathbf{M}_{ air})\\ +(\mathbf{R}\mathbf{R}^{-1}+\mathbf{R}\mathbf{J}^{-1}\mathbf{\Omega})\mathbf{R}^{-1})\hat{\mathbf{\Lambda}}_{ \text{C}}-\hat{\mathbf{\Lambda}}_{\text{C}}-\mathbf{\Phi}\end{array} \right)+\mathbf{D} \right)\] \[=\big{(}2+\tfrac{2a}{3}a\big{)}\text{diag}\Big{(}|e_{1i}|^{1+2a/3 }\Big{)}\mathbf{e}_{2}+\mathbf{\Phi}\left( \begin{array}{c}\tfrac{1}{\Theta}\bigg{(} \Theta\big{(}2+\tfrac{2a}{3}\big{)}\text{diag}\Big{(}|e_{1i}|^{1+2a/3}\Big{)} (\mathbf{I}+\mathbf{e}_{2})\\ +\tfrac{T}{2\eta T_{P}}\text{diag}\Big{(}|S_{i}|^{1-\eta/2}+3^{\eta/2}|S_{i}|^ {1+\eta/2}\Big{)}\text{sign}(\mathbf{S})-\hat{\mathbf{\Lambda}}+\mathbf{D}\bigg{)}\\ +\big{(}2+\tfrac{2a}{3}\big{)}\text{diag}\Big{(}|e_{1i}|^{1+2a/3}\Big{)} \mathbf{e}_{2}\end{array} \right)\] \[=-\big{(}2+\tfrac{2}{3}a\big{)}\text{diag}\Big{(}|e_{1i}|^{1+2a/3 }\Big{)}\Big{(}\Theta\text{sig}^{2+2a/3}(e_{1})+\Theta\text{sig}^{\frac{1+2a} {3-2a}}\big{(}\tfrac{1}{\Theta}e_{2}+\text{sig}^{2+2a/3}(e_{1})\big{)}\Big{)}\] \[\quad+\tfrac{1}{\Theta}\mathbf{\Phi}\Big{(}-\tfrac{\pi}{2\eta T_{P}} \text{diag}\Big{(}|S_{i}|^{1-\eta/2}+3^{\eta/2}|S_{i}|^{1+\eta/2}\Big{)}\text {sign}(\mathbf{S})-\hat{\mathbf{\Lambda}}+\mathbf{D}\Big{)}\] \[\quad\text{where }\mathbf{\Phi}=\tfrac{12+4a}{3-2a}\text{diag} \bigg{(}|\tfrac{e_{3i}}{\Theta}+\text{sig}^{2+2a/3}(e_{1i})|^{\frac{9+2a}{3-2 }}\Big{)},\,i=1,\,2,\,3.\] Choosing the Lyapunov function yields \[V_{S}=|S_{1}|+|S_{2}|+|S_{3}|,\] (33) where \[\mathbf{S}=[S_{1},\,\,S_{2},\,S_{3}]^{\text{T}}\]. By differentiating Equation ( 33 ), on the basis of Lemma 2, we have \[\dot{V}_{S} =|\dot{S_{1}}|+|\dot{S_{2}}|+|\dot{S_{3}}|\] (34) \[=\text{sign}(\mathbf{S})^{\text{T}}\hat{\mathbf{S}}\] \[=\text{sign}(\mathbf{S})^{\text{T}}\Bigg{(} \begin{array}{c}\tfrac{1} {\Theta}\mathbf{\Phi}\Big{(}-\tfrac{\pi}{2\eta T_{P}}\text{diag}\Big{(}|S_{i}|^{1 -\eta/2}+3^{\eta/2}|S_{i}|^{1+\eta/2}\Big{)}\text{sign}(\mathbf{S})-\hat{\mathbf{ \Lambda}}+\mathbf{D}\Big{)}\\ -\big{(}2+\tfrac{2}{3}a\big{)}\text{diag}\Big{(}|e_{1i}|^{1+2a/3}\Big{)}\mathbf{S }\end{array} \Bigg{)}\ \.\] According to Equation ( 27 ), the disturbances \[\mathbf{D}\] can be precisely estimated by PTESO when \[\overline{t}_{f}<t<T_{P}\]. Specifically, the following inequality holds. \[\dot{V}_{S} \leq-\text{min}\Big{(}\tfrac{1}{\Theta}\Phi_{ii}\Big{)}\tfrac{ \pi}{2\eta T_{P}}\Big{(}V_{S}^{1-\eta/2}+V_{S}^{1+\eta/2}\Big{)}\] \[\leq-\tfrac{1}{\Theta}\Phi_{ij}\tfrac{\pi}{2\eta T_{P}}\Big{(}V_{S }^{1-\eta/2}+V_{S}^{1+\eta/2}\Big{)}\] \[\leq G\Big{(}-\tfrac{\pi}{2\eta T_{P}}\Big{(}V_{S}^{1-\eta/2}+V_{S }^{1+\eta/2}\Big{)}\Big{)}\] where \[i=1,2,3\]. \[j=i\] when \[\Phi_{jj}=\text{min}(\Phi_{ii})\]. Then, we define \[\dot{\psi}=\tfrac{e_{2j}}{\Theta}+\text{sig}^{2+2a/3}\big{(}e_{1j} \big{)},\,G=\tfrac{1}{\Theta}\Phi_{jj}=\tfrac{12+4a}{3-2a}\tfrac{(3+2a)T_{c}}{ \sqrt{3}\pi+2\sqrt{3}\arctan\tfrac{1}{\sqrt{3}}}|\psi|^{\frac{9+2a}{3-2a}}\]. Thus, the state spaces are divided into two areas, i.e., \[\Omega_{1}=\big{\{}\big{(}e_{1j},\,\,e_{2j}\big{)}|G\geq 1\big{\}}\] and \[\overline{t}_{f}<t<T_{P}\]0.