## 3 Nonlinear Robust Control Laws
### Nonlinear Attitude Controller Based on Disturbance Observer
 \tag{43}\]  
Denoting the observer error \(\widetilde{\Delta}_{\Xi}=\Delta_{\Xi}-\widehat{\Delta}_{\Xi}\) and \(\widetilde{\epsilon}_{\xi}=\epsilon_{\xi}-\widehat{\epsilon}_{\xi}\), the error equation is given by  
\[\dot{\widetilde{\epsilon}}_{\xi}= \Xi^{T}W(\xi)-\widehat{\Xi}^{T}W(\xi)+\Delta_{\xi}-\widehat{ \Delta}_{\Xi}-A\widetilde{\epsilon}_{\xi}\] \[= \Delta_{\Xi}-\widehat{\Delta}_{\Xi}-A\widetilde{\epsilon}_{\xi}\] \[\dot{\widetilde{\Delta}}_{\Xi}= \Delta_{\Xi}-\widehat{\Delta}_{\Xi}\] \[= \Delta_{\Xi}-z_{1}-B\widetilde{\epsilon}_{\xi}\] \[\dot{\widetilde{\Delta}}_{\Xi}= \breve{\Delta}_{\Xi}-\dot{z}_{2}-B\breve{\Delta}_{\Xi}+(BA-C) \widetilde{\epsilon}_{\xi}\] \[= \breve{\Delta}_{\Xi}-D\widetilde{\epsilon}_{\xi}-B\breve{\Delta} _{\Xi}+(BA-C)\left(\widetilde{\Delta}_{\Xi}-A\widetilde{\epsilon}_{\xi}\right)\] \[= \breve{\Delta}_{\Xi}+\big{(}(C-BA)A-D\big{)}\widetilde{\epsilon}_ {\xi}+(BA-C)\widetilde{\Delta}_{\Xi}-B\breve{\Delta}_{\Xi} \tag{44}\]  
Furthermore, an error state-space equation is constructed as follows:  
\[ \begin{bmatrix}\dot{\widetilde{\epsilon}}_{\xi}\\ \dot{\widetilde{\Delta}}_{\Xi}\\ \dot{\widetilde{\Delta}}_{\Xi}\\ \dot{\widetilde{\Delta}}_{\Xi}\end{bmatrix} = \begin{bmatrix}-A&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ (C-BA)A-D&BA-C&-B&0\end{bmatrix}\begin{bmatrix}\widetilde{\widetilde{\epsilon} }_{\Xi}\\ \Delta_{\Xi}\\ \Delta_{\Xi}\end{bmatrix} + \begin{bmatrix}0\\ 0\\ 0\\ 1\end{bmatrix} \dddot{\widetilde{\Delta}}_{\Xi} \tag{45}\]  
Define \(X=[\widetilde{\epsilon}_{\xi},\widetilde{\Delta}_{\Xi},\dot{\widetilde{ \Delta}}_{\Xi},\dot{\widetilde{\Delta}}_{\Xi}]^{T}\) and  
\[\mathcal{A}= \begin{bmatrix}-A&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ (C-BA)A-D&BA-C&-B&0\end{bmatrix} \text{, }\mathcal{B}= \begin{bmatrix}0\\ 0\\ 0\\ 1\end{bmatrix}  \tag{46}\]Then, we have  
\[\dot{X}=\mathcal{A}X+\mathcal{B}\dddot{\Xi} \tag{47}\]  
Assume \(|\dddot{\Xi}|\leq\vartheta,\ \vartheta>0\) and the control law can be given by  
\[ \begin{cases}LF^{b}=JR_{\theta}^{-1}\Big{(}-\xi_{1}+R_{\theta}J^{-1}S(\dot{ \Theta})J\dot{\Theta}-\widehat{\Xi}^{T}W(\xi)+\dot{\Theta}_{d}+\dot{\xi}_{2}^{* }-\frac{1}{2\varepsilon_{3}^{2}}e_{\xi}-a_{2}e_{\xi}-\widehat{\Delta}_{\Xi} \Big{)}\\ \xi_{2}^{*}=-a_{1}\xi_{1}\\ \dot{\Xi}=W(\xi)e_{\xi}^{T}-a_{3}\widehat{\Xi}\end{cases}  \tag{48}\]  
where \(e_{\xi}=\xi_{2}-\xi_{2}^{*}\), \(a_{i},i=1,2,3\), and \(\varepsilon_{3}\) are the positive constants. Therefore, the following theorem can be obtained. Theorem 2: _For the closed-loop system of the attitude error equation (41), the state observer error equation (44), the disturbance error equation (47), and the adaptive estimate error, if there exists constants \(a_{i}>0,i=1,2,3\) and \(\varsigma>0\), and matrices \(P>0,Q>0\) such that_  
\[\mathcal{A}^{T}P+P\mathcal{A}=-Q,\ \varsigma>\frac{1}{\lambda_{min}(Q)} \tag{49}\]  
_The closed-loop system is input-to-state stable with the disturbance observer (43) and the robust adaptive controller (48). If all disturbances and uncertainties disappear, the closed-loop system is exponentially stable._  
Proof of Theorem 2: For convenience of the stability analysis, denote \(\widetilde{\varepsilon}_{\xi}=e_{\xi}-\widehat{e}_{\xi}\) and \(\widetilde{X}=\frac{X}{\xi}\) with \(\varsigma>0\) which is a small constant number.