# Robust Backstepping Control Applied to UAVs for Pest Recognition in Maize Crops
## 2 Problem Formulation and Main Results
### Main Result  
The synthesis of the proposed control algorithm is stated as follows.  
**Theorem 1**.: _Consider the system (1) and (2), under the Assumptions 1-3, for \(k>0\), and for a Lyapunov function of the form \(V_{1}(\vartheta,\bar{z})=V(\vartheta)+\frac{1}{2}\bar{z}^{2}\), with \(\bar{z}=\zeta-\mu(\vartheta)\), then, the control law_  
\[u=\frac{\partial\mu(\vartheta)}{\partial\vartheta}[f_{0}(\vartheta)+f_{1}( \vartheta)\zeta]-\frac{\partial V(\vartheta)}{\partial\vartheta}f_{1}( \vartheta)-k\bar{z}-sgn(\bar{z})\Delta \tag{4}\]  
_robustly stabilizes the system._  
**Proof of Theorem 1.** The proof is inspired on the methodology given in [26]. Adding and subtracting \(f_{1}(\vartheta)\mu(\vartheta)\) on the right-hand side of Equation (1), it yields to  
\[\dot{\vartheta} = [f_{0}(\vartheta)+f_{1}(\vartheta)\mu(\vartheta)]+f_{1}( \vartheta)[\zeta-\mu(\vartheta)] \tag{5}\] \[\dot{\zeta} = u+\delta(t,\vartheta).\]  
Define \(\overline{f_{0}}(\vartheta)=f_{0}(\vartheta)+f_{1}(\vartheta)\mu(\vartheta)\), \(\bar{z}=\zeta-\mu(\vartheta)\), then its derivative is given by \(\dot{\bar{z}}=\dot{\zeta}-\dot{\mu}(\vartheta)=(u+\delta(t,\Now, the system (5) is expressed in the classical backstepping form  
\[\dot{\vartheta} = \overline{f_{0}}(\vartheta)+f_{1}(\vartheta)\tilde{z} \tag{8}\] \[\dot{\tilde{z}} = \vartheta+\delta(t,\vartheta),\]  
when \(\tilde{z}=0\), it guarantees that the system (8) has an equilibrium point at the origin (according to Assumption 2).  
A positive definite Lyapunov candidate function of the form \(V_{1}(\vartheta,\tilde{z})=V(\vartheta)+\frac{1}{2}\tilde{z}^{2}\) is proposed, and its derivative along the trajectories of system (8), according to Assumption 2, is given by  
\[\left.\dot{V}_{1}(\vartheta,\tilde{z})\right|_{(\ref{eq:1})} = \frac{\partial V(\vartheta)}{\partial\vartheta}\overline{f_{0}} (\vartheta)+\frac{\partial V(\vartheta)}{\partial\vartheta}f_{1}(\vartheta) \tilde{z}+z\dot{\tilde{z}}\] \[\leq -W(\vartheta)+\frac{\partial V(\vartheta)}{\partial\vartheta}f_{ 1}(\vartheta)\tilde{z}+z\dot{\tilde{z}},\]  
where \(\tilde{z}=\bar{v}+\delta(t,\vartheta)\), set  
\[\bar{v}=-k\tilde{z}-\frac{\partial V(\vartheta)}{\partial\vartheta}f_{1}( \vartheta)-sgn(\tilde{z})\Delta, \tag{9}\]  
with \(k>0\); substituting into the previous equation, it yields  
\[\left.\dot{V}_{1}(\vartheta,\tilde{z})\right|_{(\ref{eq:1})}\leq-W(\vartheta) -k\tilde{z}^{2}+z\delta(t,\vartheta)-\tilde{z}sgn(\tilde{z})\Delta. \tag{10}\]  
Majorizing the term  
\[\tilde{z}\delta(t,\vartheta)\leq|\tilde{z}\delta(t,\vartheta)|\leq|\tilde{z}|| \delta(t,\vartheta)|\leq|\tilde{z}|\Delta,\]  
and \(|\tilde{z}|\triangleq zsgn(\tilde{z})\); so, \(|\tilde{z}|\Delta=zsgn(\tilde{z})\Delta\); substituting into Equation (10), it yields  
\[\left.V_{1}(\vartheta,\tilde{z})\right|_{(\ref{eq:1})}\leq-W(\vartheta)-k \tilde{z}^{2}<0,\]  
so, the origin of the system (8) is robustly stable; since \(\mu(0)=0\), the origin of system (1) and (2) is robustly stable too. The input \(u=\vartheta+\hat{\mu}(\vartheta)\) is obtained from Equation (6), and replacing Equations (7) and (9) into \(u\), the result follows in Equation (4).  
The control action (4) is used to solve the trajectory tracking problem of a quadrotor whose model is shown in the following section.