## Appendix A  
Proof of Equation (11):  
First, using Equation (7), we can get:  
\[\widetilde{x}_{2j}=\alpha_{3j}^{-3}(\alpha_{3j}^{2}\widetilde{x}_{3j}+ \dot{\widetilde{x}}_{3j})-\frac{\dot{\Psi}}{\alpha^{4}{}_{3j}} \tag{13}\]  
Because the estimated perturbation is defined in Equation (6) as:  
\[\dot{\psi}_{j}=\alpha_{3j}\Big{(}\alpha_{3}\hat{x}_{2j}-\hat{x}_{3j}\Big{)} \tag{14}\]  
\(\widetilde{x}_{3j}\) is simplified as:  
\[\widetilde{x}_{3j}=\hat{x}_{3j}-\alpha_{3j}x_{2j}+\frac{\Psi_{j}}{\alpha_{3j}} \tag{15}\]  
\[=\hat{x}_{3j}-\alpha_{3j}x_{2j}+(\frac{\Psi_{j}-\widetilde{\Psi}}{\alpha_{3j} })\] \[=-\alpha_{3j}\widetilde{\psi}+\alpha_{3j}\widetilde{x}_{2j}\]  
Since SMCSPO is discussed in the manifold \(\left|\hat{\psi}_{j}\right|\leq\varepsilon_{cj}\), from Equation (7) we can get:  
\[\dot{\widetilde{x}}_{1j}=\widetilde{x}_{2j}-\frac{k_{1j}}{\varepsilon_{0j}} \widetilde{x}_{1j} \tag{16}\]  
\[\widetilde{x}_{1j}=\frac{(-\ddot{\widetilde{x}}_{2j}+\widetilde{\Psi}_{j})}{ \frac{k_{2j}}{\varepsilon_{0j}}} \tag{17}\]  
\[\dot{\widetilde{x}}_{2j}+\frac{k_{2j}}{\varepsilon_{0j}}\widetilde{x}_{1j}= \widetilde{\Psi}_{j} \tag{18}\]  
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