# Trajectory Tracking Outer Loop Regressor-Based Adaptive Controller for a Quadrotor
## 3 Main Result
### Convergence Analysis  
The closed-loop system resulting from the dynamics of the quadrotor with an inner controller (6), the outer loop regressor-based adaptive controller (17)-(18), the update law (19), and the change in variable in (21)-(22) is given by  
\[ \begin{array}{rcl}\frac{d}{dt}\mathbf{e}&=&-\Gamma\mathbf{e}+\mathbf{s},\\ \Lambda_{1}\frac{d}{dt}\mathbf{s}&=&-K_{\mathrm{s}}\mathbf{s}-K_{\mathrm{i}}\bar{\mathbf{ \xi}}+\Phi(\bar{\mathbf{q}}_{d},\dot{\mathbf{q}}_{d},\dot{\mathbf{e}})\bar{\mathbf{\Theta}},\\ \frac{d}{dt}\bar{\mathbf{\xi}}&=&\mathbf{s},\\ \frac{d}{dt}\bar{\mathbf{\Theta}}&=&-N\Phi(\bar{\mathbf{q}}_{d},\dot{\mathbf{q}}_{d},\dot{ \mathbf{e}})^{\top}\mathbf{s}.\end{array}  \tag{23}\]  
Notice that the origin \(\mathbf{\varrho}=[\mathbf{e}^{\top}\mathbf{s}^{\top}\mathbf{\bar{\mathbf{\xi}}}^{\top}\mathbf{\bar{ \mathbf{\Theta}}}^{\top}]^{\top}=\mathbf{0}\in\mathbb{R}^{20}\) is an equilibrium point of the overall closed-loop system (23). **Proposition 1**. : _Assume that matrices \(\Gamma\) and \(K_{\mathrm{s}}\) are selected such that the inequality_  
\[\lambda_{\min}\{\Gamma\}>\frac{1}{4\lambda_{\min}\{K_{\mathrm{s}}\}} \tag{24}\]  
_is satisfied._  
_Then, the equilibrium point \(\mathbf{\varrho}=\mathbf{0}\) is uniformly stable in the sense of Lyapunov. Moreover, the signals \(\bar{\mathbf{\Theta}}(t)\) and \(\bar{\mathbf{\xi}}(t)\) are bounded, \(\forall t\geq t_{0}\), and the limit (12) is fulfilled._  
Proof. : Consider the following Lyapunov function candidate  
\[V(\mathbf{\varrho})=\frac{1}{2}\mathbf{e}^{\top}\mathbf{e}+\frac{1}{2}\mathbf{s}^{\top} \Lambda_{1}\mathbf{s}+\frac{1}{2}\mathbf{\bar{\xi}}^{\top}K_{\mathrm{i}}\bar{\mathbf{\xi} }+\frac{1}{2}\bar{\mathbf{\Theta}}^{\top}N^{-1}\bar{\mathbf{\Theta}}, \tag{25}\]  
which is globally positive definite, radially unbounded, and decrescent. The time derivative of (25) is given by  
\[\dot{V}=\mathbf{e}^{\top}\dot{\mathbf{e}}+\mathbf{s}^{\top}\Lambda_{1}\dot{\mathbf{s}}+\bar\mathbf{{\xi}}^{\top}K_{\mathrm{i}}\dot{\bar{\mathbf{\xi}}}+\bar{\mathbf{\Theta}}^{\top}N^{-1} \dot{\bar{\mathbf{\Theta}}}. \tag{26}\]  
By substituting the closed-loop dynamics in (23) into (26) and using the fact that  
\[\mathbf{s}^{\top}\Phi(\bar{\mathbf{q}}_{d},\dot{\mathbf{q}}_{d},\dot{\mathbf{e}})\bar{\mathbf{ \Theta}}=\bar{\mathbf{\Theta}}^{\top}\Phi(\bar{\mathbf{q}}_{d},\dot{\mathbf{q}}_{d},\dot{ \mathbf{e}})^{\top}\mathbf{s},\]  
the following is obtained:  
\[\dot{V}(\mathbf{e},\mathbf{s})=-\mathbf{e}^{\top}\Gamma\mathbf{e}+\mathbf{e}^{\top}\mathbf{s}-\mathbf{s}^{ \top}K_{\mathrm{s}}\mathbf{s},\]  
which can be rewritten as  
\[\dot{V}(\mathbf{\chi})=-\mathbf{\chi}^{\top}Q\mathbf{\chi}, \tag{27}\]  
where \(\mathbf{\chi}=[\mathbf{e}^{\top}\mathbf{s}^{\top}]^{\top}\in\mathbb{R}^{8}\) and  
\[Q= \begin{bmatrix}\Gamma&-\frac{1}{2}I_{4}\\ -\frac{1}{2}I_{4}&K_{\mathrm{s}}\end{bmatrix} \in\mathbb{R}^{8\times 8},\]where \(I_{4}\in\mathbb{R}^{4\times 4}\) is the identity matrix. Notice that (27) is always a negative function if \(Q\) is a positive definite matrix. In agreement with the [PERSON] complement [27], \(Q\) is a positive definite matrix if \(\Gamma>0\) and \(K_{s}-\frac{1}{4}\Gamma^{-1}>0\). This condition is satisfied with \(\Gamma\) selected as in (24). Therefore, by fulfilling (24), \(\dot{V}(\mathbf{\chi})\) is guaranteed to be a negative semidefinite function. Based on [PERSON]'s method [28], we have that \(\varrho=\mathbf{0}\) is a uniformly stable equilibrium point of the overall closed-loop system (23), which implies that \(\varrho(t)\), \(\dot{\varrho}(t)\in L^{20}_{\infty}\).