# A Model-Free Online Learning Control for Attitude Tracking of Quadrotors
## 3 Model-Free Online Learning Control Design
### Stability Analysis  
Substituting (8)-(10) into (7) and combining the result with (6) yields the dynamical equation of attitude synthesis error:  
\[\mathbf{J}\mathbf{\dot{\varepsilon}}=-\mathbf{\dot{\varepsilon}}^{\wedge}\mathbf{J}\mathbf{\omega}+ \mathbf{S}\mathbf{\omega}_{e}+\mathbf{\beta}-\mathbf{K}\mathbf{\varepsilon} \tag{12}\]  
where  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]  
\[\mathbf{\beta}=k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{I}\mathbf{\Omega}_{d}-\mathbf{\Omega}_{d}^{ \wedge}\mathbf{I}\mathbf{\Omega}_{d}-\mathbf{J}\mathbf{\dot{\Omega}}_{d}+\mathbf{d}+\mathbf{L}(t) \tag{14}\]  
The following lemma is given as a preparation for the proof of the MFOLC's stability. Under Assumptions 1-3, the following inequalities hold:  
\[\|\mathbf{S}\mathbf{\omega}_{e}\|\leq\sigma\|\mathbf{\dot{\varepsilon}}\| \tag{15}\]  
\[\|\mathbf{\beta}\|\leq\rho \tag{16}\]  
where  
\[\sigma=\overline{J}\big{(}1.5k_{p}+3\overline{\omega}_{d}\big{)} \tag{17}\]  
\[\rho=\overline{J}\big{(}\overline{\omega}_{d}^{2}+k_{p}\overline{\omega}_{d}+ \overline{\omega}_{d}\big{)}+\overline{d}+\overline{k}_{l}\overline{u} \tag{18}\]  
Proof of Lemma 1.: Note that \(\|\mathbf{a}^{\wedge}\|=\|\mathbf{a}\|\) for \(\mathbf{a}\in\mathbb{R}^{3\times 1}\), \(\|s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\|=1\), and \(\|\mathbf{R}(\mathbf{q}_{e})\|=1\). According to the Minkowski inequality and Assumptions 1 and 3, we have  
\[\|\mathbf{S}\|\leq\|\mathbf{J}\|\bigg{(}\frac{1}{2}k_{p}\big{\|}s_{e}\mathbf{I}_{3}+\mathbf{v} _{e}^{\wedge}\big{\|}+k_{p}\|\mathbf{v}_{e}\|+3\|\mathbf{\Omega}_{d}\|\bigg{)}\ \leq\ \overline{J}\big{(}1.5k_{p}+3\overline{\omega}_{d}\big{)}=\sigma \tag{19}\]  
It is clear that \(\sigma\geq 0\). Recall that since \(\|\mathbf{\dot{\varepsilon}}\|\leq k_{p}+\|\mathbf{\omega}_{e}\|\) and \(k_{p}>0\), \(\|\mathbf{\omega}_{e}\|<\|\mathbf{\dot{\varepsilon}}\|\). Therefore,  
\[\|\mathbf{S}\mathbf{\omega}_{e}\|\leq\|\mathbf{S}\|\|\mathbf{\omega}_{e}\|\leq\sigma\|\mathbf{ \dot{\varepsilon}}\| \tag{20}\]  
Then Equation (15) holds. Similarly, according to the Minkowski inequality and Assumption 1,  
\[\|\mathbf{L}\mathbf{u}(t-\tau)\|\leq\|\mathbf{L}\|\|\mathbf{u}(t-\tau)\|\leq\overline{k}_{l} \overline{u} \tag{21}\]  
holds, where \(\overline{k}_{l}=\max(k_{l,1},k_{l,2},k_{l,3})\). Upon combining Assumptions 2 and 3, we can conclude that  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]0  
As such, \(\rho>0\). Therefore, this completes the proof. \(\square\)  
The stability analysis of the closed-loop attitude system can be stated in the following Theorem. Consider the quadrotor attitude system governed by (2) and (3) under Assumptions 1-3. With the application of the control law (8)-(10), we suppose that the control parameters are chosen so that  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]1  
where \(\underline{k}=min(k_{1},k_{2},k_{3})\), holds. Then, the attitude synthesis error \(\mathbf{\dot{\varepsilon}}\) is UIUB.