# A Model-Free Online Learning Control for Attitude Tracking of Quadrotors
## 3 Model-Free Online Learning Control Design
### Stability Analysis
Proof of Theorem 1.: Consider the following Lyapunov function candidate:  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]2Obviously, \(V\geq 0\) and \(V=0\) if and only if \(\mathbf{\epsilon}=[0,0,0]^{T}\). The derivative of \(V\) is  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]3  
Substituting (12) into (25) yields  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]4  
Note that \(\mathbf{\epsilon}^{T}\mathbf{\epsilon}^{\wedge}\mathbf{J}\mathbf{\omega}=0\), and according to Lemma 1, the following inequality holds:  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]5  
Since \(\|\mathbf{\epsilon}\|\geq 0\) and \(\rho>0\), we have  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]6  
Substituting (28) into (27) yields  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]7  
where \(\pi=\underline{k}-\sigma-1\), and \(\delta=\frac{\rho^{2}}{4}\). It is seen from (25) that \(\dot{V}<0\) when \(\mathbf{\epsilon}\) are outside of set \(\mathcal{D}=\left\{\mathbf{\epsilon}\big{|}\|\mathbf{\epsilon}\|\leq\frac{\rho}{2\sqrt {k}-\sigma-1}\right\}\). This implies that \(V(t)\) decreases monotonically outside set \(\mathcal{D}\). Hence, the attitude synthesis error in the closed-loop system is bounded. Moreover, we can choose a sufficiently small \(\varepsilon_{0}>0\), let \(\epsilon^{\prime}=\varepsilon_{0}+\frac{\rho}{2\sqrt{k}-\sigma-1}\), and define set \(\mathcal{D}^{\prime}=\left\{\mathbf{\epsilon}\big{|}\|\mathbf{\epsilon}\|\leq\epsilon^ {\prime}\right\}\) to guarantee that  
\[\mathbf{S}=\frac{1}{2}k_{p}\mathbf{J}\big{(}s_{e}\mathbf{I}_{3}+\mathbf{v}_{e}^{\wedge}\big{)}+ k_{p}\mathbf{v}_{e}^{\wedge}\mathbf{J}+\big{(}\mathbf{I}\mathbf{\Omega}_{d}\big{)}^{\wedge}- \mathbf{\Omega}_{d}^{\wedge}\mathbf{J}-\mathbf{J}\mathbf{\Omega}_{d}^{\wedge} \tag{13}\]8  
It can be concluded from (30) that there exists a \(T(\epsilon^{\prime})>0\), such that \(\|\mathbf{\epsilon}(t)\|\leq\epsilon^{\prime}\) for \(t\geq T(\epsilon^{\prime})\). This shows that \(\mathbf{\epsilon}\) is UUB from its definition [24]. According to Theorem 2 in [25], when \(\epsilon\) converges to a small region \(\epsilon^{\prime}\), \(|\omega_{\epsilon,i}(t)|\leq 2\epsilon^{\prime}\), \(|q_{\epsilon,i}(t)|\leq\epsilon^{\prime}/k_{p}\), \(i=1,2,3\). This completes the proof. \(\square\)  
**Remark 2**. : _The proposed MFOLC is more intuitive than the ordinary OLC in control parameter tuning. The following steps can be used for parameter tuning in the MFOLC. Firstly, one can set an appropriate \(\overline{\omega}_{d}\) according to the quadrotor's task and estimate \(\overline{J}\) through the weight and size of the quadrotor.