## 3 Design of Controller and Nonlinear Disturbance Observer
### Design of the Disturbance Observer  
**Assumption 1**. : _The disturbance \(|\mathbf{\rho}(t)|\) is bounded such that there exists an unknown constant that satisfies \(|\mathbf{\rho}(t)|\leq\overline{\mathbf{p}}\)._  
**Theorem 4**. : _The disturbances \(\mathbf{\rho}=\big{[}\rho_{2}\ \rho_{4}\ \rho_{6}\ \rho_{8}\big{]}^{T}\) are estimated by the following novel nonlinear disturbance observers:_  
\[ \begin{array}{c}\dot{\hat{\mathbf{\Theta}}}=\mu_{2}[\mathbf{f}(\mathbf{x})+\mathbf{g}\mathbf{u}- \chi_{2}-\check{\mathbf{\xi}}_{2}-\hat{\mathbf{\rho}}],\\ \dot{\hat{\mathbf{\rho}}}=-\mathbf{\Lambda}^{2}[\eta_{\rho 1}\mathrm{sinh}(c_{\rho 1} \mathbf{\varphi}_{1}(\widehat{\mathbf{\Theta}}))+\eta_{\rho 2}\mathrm{sinh}(c_{\rho 2} \mathbf{\Lambda}^{-1}\mathbf{\varphi}_{2}(\hat{\mathbf{\rho}}))],\end{array}  \tag{78}\]  
_where \(\mathbf{\Theta}=[\zeta_{21}\ \zeta_{41}\ \zeta_{61}\ \zeta_{81}]^{T}\) denotes the state vector, including the super-twisting state vector, \(\mathbf{\mu}_{2}=diag(\mu_{2},\mu_{4},\ \mu_{6},\ \mu_{8})\), \(\chi_{2}=\big{[}\chi_{21}\ \chi_{22}\ \chi_{52}\ \chi_{72}\big{]}^{T}\) denotes the state vector of the tracking differentiator, \(\check{\xi}_{2}=[\check{\xi}_{12}\ \check{\xi}_{32}\ \check{\xi}_{52}\ \check{\xi}_{72}]^{T}\) denotes the state vector of the compensating signal, \(\mathbf{\varphi}_{1}(\widehat{\mathbf{\Theta}})=\)\(\mathrm{x}s\check{\mathbf{\Theta}}^{1/2}(\widehat{\mathbf{\Theta}})+(1-\kappa)sig^{(2+ \gamma)/2}(\widehat{\mathbf{\Theta}})\), \(\mathbf{\varphi}_{2}(\hat{\mathbf{\rho}})=sig^{1/2}(\hat{\mathbf{\rho}})\), \(\widetilde{\mathbf{\Theta}}=\mathbf{\Theta}-\hat{\mathbf{\Theta}}\), \(\widetilde{\mathbf{\rho}}=\mathbf{\rho}-\hat{\mathbf{\rho}}\), \(\widehat{\mathbf{\Theta}}\) and \(\hat{\mathbf{\rho}}\) denote the estimates of \(\mathbf{\Theta}\) and \(\mathbf{\rho}\), \(\mathbf{\Lambda}\), \(\mathbf{\eta}_{\rho 1}\), \(\mathbf{\eta}_{\rho 2}\),\(c_{\rho 1}\), and \(c_{\rho 1}\) are diagonal positive constant matrices, respectively. Evidently, \(\check{\mathbf{\Theta}}\to\mathbf{\Theta}\) and \(\hat{\mathbf{\rho}}\to\mathbf{\rho}\) when \(\mathbf{\Lambda}\to\infty\)._  
Proof. : When \(\mathbf{\Lambda}\to\infty\), \(\Big{|}\dot{\hat{\mathbf{\rho}}}\Big{|}=\Big{|}\mathbf{\Lambda}^{2}[\mathbf{\eta}_{\rho 1} \mathrm{sinh}(c_{\rho 1}\mathbf{\varphi}_{1}(\widehat{\mathbf{\Theta}}))+\mathbf{\eta}_{\rho 2} \mathrm{sinh}(c_{\rho 2}\mathbf{\Lambda}^{-1}\mathbf{\varphi}_{2}(\hat{\mathbf{\rho}}))] \Big{|}\) can be assumed as an infinitely large value. This indicates that the variation in \(\hat{\mathbf{\rho}}\) significantly exceeds \(\mathbf{f}(\mathbf{x})+\mathbf{g}\mathbf{u}-\chi_{2}-\check{\xi}_{2}\). Furthermore, it suggests that \(\lim\limits_{\mathbf{\Lambda}\to\infty}\frac{d(\mathbf{f}(\mathbf{x})+\mathbf{g}\mathbf{u}-\chi_{ 2}-\check{\xi}_{2}+\hat{\rho})}{dt}\)\(=\)\(\hat{\mathbf{\rho}}\), \(\lim\limits_{\mathbf{\Lambda}\to\infty}\mathbf{\Lambda}^{-1}(\mathbf{f}(\mathbf{x})+\mathbf{g}\mathbf{u }-\chi_{2}-\check{\xi}_{2}+\hat{\rho})\)\(=\)\(\mathbf{\Lambda}^{-1}\hat{\mathbf{\rho}}\). Hence, it is easy to note that (78) holds according to Theorem (1) when \(\mathbf{\mu}_{2}[\mathbf{f}(\mathbf{x})+\mathbf{g}\mathbf{u}-\chi_{2}-\check{\xi}_{2}+\hat{\mathbf{ \rho}}]\) is considered as \(\hat{\mathbf{\Theta}}\). This concludes the proof. To compare of the proposed observer with the conventional observers, two disturbance observers were provided. The first observer is designed using the structure of the tracking differentiator proposed by [PERSON] et al. [28; 29] expressed in (20) as follows:  
\[