## 3 Nonlinear Robust Control Laws
### Nonlinear Attitude Controller Based on Disturbance Observer
 \tag{57}\]  
into (55), equation (58) is rewritten as  
\[\dot{V}\leq -a_{1}\xi_{1}^{T}\xi_{1}-a_{2}\epsilon_{\xi}^{T}\epsilon_{\xi}+ \epsilon_{\xi}^{T}\Delta\Xi^{T}W(\xi)-\varsigma\lambda_{min}(Q)\widetilde{X}^ {T}\widetilde{X}+\widetilde{X}^{T}\widetilde{X}+\|P\|^{2}\|\widetilde{\Delta }_{\Xi}\|^{2}-tr(\Delta\Xi^{T}\Delta\hat{\bar{\Xi}})+\frac{\epsilon_{3}^{2}}{ 2}\epsilon_{\Delta}^{T}\epsilon_{\Delta} \tag{58}\]  
Because  
\[e_{\xi}^{T}\Delta\Xi^{T}W(\xi)=tr\Big{(}\Delta\Xi^{T}W(\xi)\epsilon_{\xi}^{T} \Big{)} \tag{59}\]  
Thus,  
\[\dot{V}\leq -a_{1}\xi_{1}^{T}\xi_{1}-a_{2}\epsilon_{\xi}^{T}\epsilon_{\xi}-( \varsigma\lambda_{min}(Q)-1)\widetilde{X}^{T}\widetilde{X}+tr\Big{(}\Delta\Xi^ {T}W(\xi)\epsilon_{\xi}^{T}\Big{)}-tr(\Delta\Xi^{T}\Delta\hat{\bar{\Xi}})+\|P \|^{2}\|\widetilde{\Delta}_{\Xi}\|^{2}+\frac{\epsilon_{3}^{2}}{2}\epsilon_{ \Delta}^{T}\epsilon_{\Delta} \tag{60}\]  
Use the adaptive law \(\Delta\hat{\bar{\Xi}}\), and we obtain  
\[\dot{V}\leq -a_{1}\xi_{1}^{T}\xi_{1}-a_{2}\epsilon_{\xi}^{T}\epsilon_{\xi}-( \varsigma\lambda_{min}(Q)-1)\widetilde{X}^{T}\widetilde{X}+tr\Big{(}\Delta\Xi^ {T}W(\xi)\epsilon_{\xi}^{T}\Big{)}-tr\Big{(}\Delta\Xi^{T}\Big{(}W(\xi) \epsilon_{\xi}^{T}-a_{3}\bar{\Xi}\Big{)}\Big{)}+\|P\|^{2}\|\widetilde{\Delta} _{\Xi}\|^{2}+\frac{\epsilon_{3}^{2}}{2}\epsilon_{\Delta}^{T}\epsilon_{\Delta}\] \[\leq -a_{1}\xi_{1}^{T}\xi_{1}-a_{2}\epsilon_{\xi}^{T}\epsilon_{\xi}-( \varsigma\lambda_{min}(Q)-1)\widetilde{X}^{T}\widetilde{X}+a_{3}tr\Big{(}\Delta \Xi^{T}\bar{\Xi}\Big{)}+\|P\|^{2}\|\widetilde{\Delta}_{\Xi}\|^{2}+\frac{ \epsilon_{3}^{2}}{2}\epsilon_{\Delta}^{T}\epsilon_{\Delta} \tag{61}\]  
Term \(a_{3}tr\Big{(}\Delta\[PERSON]^{T}\bar{\[PERSON]}\Big{)}\) can be rewritten as  
\[a_{3}tr\Big{(}\Delta\[PERSON]^{T}\bar{\[PERSON]}\Big{)}= -\frac{a_{3}}{2}tr\bigg{(}\Big{(}\[PERSON]-\bar{\[PERSON]}\Big{)}^{T}\Big{(} \[PERSON]-\bar{\[PERSON]}\Big{)}\bigg{)}+\frac{a_{3}}{2}tr\Big{(}\[PERSON]^{T}\bar{\[PERSON]}\Big{)}- \frac{a_{3}}{2}tr\Big{(}\bar{\[PERSON]}^{T}\bar{\[PERSON]}\Big{)}\] \[\leq -\frac{a_{3}}{2}tr\bigg{(}\Big{(}\[PERSON]-\bar{\[PERSON]}\Big{)}^{T}\Big{(} \[PERSON]-\bar{\[PERSON]}\Big{)}\bigg{)}+\frac{a_{3}}{2}tr\Big{(}\[PERSON]^{T}\bar{\[PERSON]}\Big{)}\] \[\leq -\frac{a_{3}}{2}tr\Big{(}\Delta\[PERSON]^{T}\Delta\[PERSON]\Big{)}+\frac{a_{3}}{ 2}tr\Big{(}\[PERSON]^{T}\bar{\[PERSON]}\Big{)} \tag{62}\]Because  
\[\lambda_{min}(P)\|\widetilde{X}\|^{2}\leq\widetilde{X}^{T}P\widetilde{X}\leq \lambda_{max}(P)\|\widetilde{X}\|^{2} \tag{63}\]  
Then, we have  
\[ \begin{split}\dot{V}&\leq-a_{1}\xi_{1}^{T}\xi_{1}-a_{2 }e_{\xi}^{T}e_{\xi}-\frac{(\varepsilon\lambda_{min}(Q)-1)}{\lambda_{max}(P)} \widetilde{X}^{T}P\widetilde{X}-\frac{a_{3}}{2}tr\bigg{(}\Delta\Xi^{T}\Delta \Xi\bigg{)}+\|P\|^{2}\|\overset{\cdot}{\Delta\Xi}\|^{2}+\frac{a_{3}}{2}tr \Big{(}\Xi^{T}\Xi\Big{)}+\frac{\varepsilon_{3}^{2}}{2}e_{\Delta}^{T}e_{\Delta} \\ &\leq-2\mu V+\frac{t}{2}\|\Delta\Xi\|^{2}\\ &\text{where }\xi>\frac{1}{\lambda_{min}(Q)},\iota=\max\{a_{3}, \varepsilon_{3}^{2},2\},\text{and}\\ &\mu=\min\Bigl{\{}a_{1},a_{2},(\varepsilon\lambda_{min}(Q)-1)/ \lambda_{max}(P),\frac{a_{3}}{2}\Bigr{\}}\\ &\|\Delta_{\omega}\|^{2}=\|P\|^{2}\|\overset{\cdot}{\Delta\Xi}\|^ {2}+tr\Big{(}\Xi^{T}\Xi\Big{)}+e_{\Delta}^{T}e_{\Delta}\end{split}  \tag{64}\]  
With the integral on both sides of inequality (64), then we have  
\[V\leq e^{-2\mu t}V(0)+\frac{t}{4\mu}\Big{(}1-e^{-2\mu t}\Big{)}\left(\sup_{0 \leq\tau\leq t}\|\Delta_{\omega}\|\right) \tag{65}\]  
Define \(\chi=[\xi_{1},e_{\xi},\Delta\Xi,\widetilde{X}]^{T}\), and we have  
\[\|\chi\|\leq e^{-\mu t}\|\chi(0)\|+\sqrt{\frac{t}{2\mu}(1-e^{-\mu t})}\left( \sup_{0\leq\tau\leq t}\|\Delta_{\omega}\|\right) \tag{66}\]  
where \(\chi(0)=[\xi_{1}(0),e_{\xi}(0),\Delta\Xi(0),\widetilde{X}(0)]^{T}\) and \(\Delta\Xi(0)=\Xi(0)-\widetilde{\Xi}(0),\widetilde{\Xi}(0)>0\). Therefore, the closed-loop system is ISS. Moreover, if the uncertainty does not exist, that is, \(\Delta_{\omega}=0\), the closed-loop system is exponentially stable.