## 3 Adaptive Robust Control Design
### Disturbance Observer Design  
To actively compensate for the effect of unmodeled disturbances, uncertain nonlinearities, external load, and the imperfection in approximating unknown continuous functions in system dynamics (11) on the control system, a pair of NN-based disturbance observers are proposed as  
\[ \begin{split}\dot{d}_{1}&=\frac{1}{\kappa_{1}}\left( \tilde{x}_{2}-x_{3}-\hat{f}_{1}-\hat{d}_{1}\right)\\ \dot{\hat{d}}_{2}&=\frac{1}{\kappa_{2}}\left( \tilde{x}_{3}-\hat{f}_{2}-\hat{g}_{2}u-\hat{G}_{1}-\hat{d}_{2}\right)\end{split}  \tag{37}\]  
where \(\kappa_{1}\) and \(\kappa_{2}\) are small positive constants, \(\hat{d}_{1}\) and \(\hat{d}_{2}\) are estimates of \(d_{1}\) and \(d_{2}\), respectively. \(\hat{G}_{1}\) is the NN-based approximation of \(G_{1}=\hat{g}_{2}u\), which is generated by the NN weighting approximation error of \(g_{2}\). The function \(G_{1}\) is represented by RBF NN as  
\[G_{1}=\mathbf{W}_{G_{1}}^{\mathrm{T}}\mathbf{h}_{G_{1}}(\mathbf{\mathcal{O}})+ \varepsilon_{G_{1}} \tag{38}\]  
where the weight value \(\|\mathbf{W}_{G_{1}}\|\) and approximation error \(\varepsilon_{G_{1}}\) are bounded by positive constants; i.e., \(\|\mathbf{W}_{G_{1}}\|\leq W_{G_{1}M}\), \(|\varepsilon_{G_{1}}|\leq\varepsilon_{G_{1}M}\). From (37), the approximated value \(\tilde{G}_{1}\) and the approximation error \(\tilde{G}_{1}\) are given by  
\[ \begin{split}\tilde{G}_{1}&=\tilde{\mathbf{W}}_{G_ {1}}^{\mathrm{T}}\mathbf{h}_{G_{1}}\\ \tilde{G}_{1}&=\tilde{\mathbf{W}}_{G_{1}}^{\mathrm{T }}\mathbf{h}_{G_{1}}+\varepsilon_{G_{1}}\end{split}  \tag{39}\]  
where \(\tilde{\mathbf{W}}_{G_{1}}=\mathbf{W}_{G_{1}}-\hat{\mathbf{W}}_{G_{1}}\) and the weight is updated according to the following law:  
\[\hat{\mathbf{W}}_{G_{1}}=\frac{1}{2}\gamma_{G_{1}}\sigma\mathbf{h}_{G_{1}}- \Gamma_{G_{1}}\hat{\mathbf{W}}_{G_{1}} \tag{40}\]  
where \(\gamma_{G_{1}}\) and \(\Gamma_{G_{1}}\) are positive constants. The disturbance estimation errors are defined by  
\[ \begin{split}\tilde{d}_{1}&=d_{1}-\hat{d}_{1}\\ \tilde{d}_{2}&=d_{2}-\hat{d}_{2}\end{split}  \tag{41}\]  
Assuming that the state derivatives are perfectly calculated, based on (11) and (37), the disturbance estimation error dynamics are obtained as  
\[ \begin{split}\dot{\hat{d}}_{1}&=-\frac{1}{\kappa_{1 }}\big{(}\tilde{f}_{1}+\tilde{d}_{1}\big{)}+\hat{d}_{1}\\ \dot{\hat{d}}_{2}&=-\frac{1}{\kappa_{2}}\big{(}\tilde {f}_{2}+\tilde{G}_{1}+\tilde{d}_{2}\big{)}+\hat{d}_{2}\end{split}  \tag{42}\]  
**Remark 2**. : _With the help of the exact high-order differentiators (27) and (28), and when the system dynamics are assumed to be known--i.e., \(\tilde{f}_{1}=0\), \(\tilde{f}_{2}=0\), and \(\tilde{g}_{2}=0\)--the disturbance observers (37) guarantee the uniformly asymptotic stability under the constant disturbances and boundedness stability under the time-varying disturbances._Based on (19)-(21), (27), (28), (33), and (37), the control laws are redesigned as  
\[ \begin{split} u_{eq}&=\frac{1}{\hat{\delta}_{2}} \begin{pmatrix}-\hat{f}_{2}-\frac{1}{2}\hat{G}_{1}-\frac{1}{2}\hat{G}_{2}-\hat{ d}_{2}-\hat{f}_{1}+\hat{d}_{1}\\ +x_{1d}^{(3)}&-(\lambda+k)\left(x_{3}+\hat{f}_{1}+\hat{d}_{1}-\check{x}_{1d} \right)\\ -k\lambda(\check{x}_{2}-\hat{x}_{1d})\end{pmatrix}\\ u_{sw}&=-\frac{1}{\hat{\delta}_{2}}[\gamma_{1}\text{sgn}(\sigma)+\eta_{2 }\sigma]\\ u&=u_{eq}+u_{sw}\end{split}  \tag{43}\]  
where  
\[ \begin{split} e_{1}=x_{1}-x_{1d};e_{2}=\check{x}_{2}-\hat{x}_{1d} \\ s=e_{2}+\lambda e_{1};\sigma=\check{s}+ks\end{split}  \tag{44}\]  
and the function \(G_{2}=\varepsilon_{g_{2}}u\) is approximated by adopting RBF NN as  
\[G_{2}=\mathbf{W}_{G_{2}}^{\mathrm{T}}\mathbf{h}_{G_{2}}(\mathbf{\Theta})+ \varepsilon_{G_{2}} \tag{45}\]  
with \(\|\mathbf{W}_{G_{2}}\|\leq W_{G_{2}M}\) and \(|\varepsilon_{G_{2}}|\leq\varepsilon_{G_{2}M}\).