## 2 Literature Review
### Theoretical Analysis of the Proposed Results: Lyapunov Stability Criteria for Agricultural Drones  
Drone systems for agriculture are among the many dynamical systems whose stability can be examined using the robust Lyapunov stability criterion. This criterion is utilized in control system design to guarantee that the agricultural drone system is stable throughout operation. We assume the difference between \(y\) and the steady state. If that reaches zero, then the state space is stable. If it increases, and if \(y\) moves further away from the steady state, it is unstable. We pick a (non-negative) function such that the value decreases along the trajectory of the agricultural drone. Consider a mass (\(m\)) acted on by a dissipative force as  
\[m\ddot{x}=-b\dot{x} \tag{55}\]  
where \(m\) is mass of the agricultural drone and \(b\) is the air drag friction coefficient.  
This implies that  
\[m\dot{v}=-bv \tag{56}\]  
where \(v\) is the speed of the agricultural drone.  
\(X=0,v=0\) at equilibrium points.  
The kinetic energy could be  
\[E(t)=\frac{1}{2}mv^{2} \tag{57}\]and its derivative could be  
\[\dot{E}(t)=mv.\dot{v}=-bv^{2}<0 \tag{58}\]  
Since \(E(t)\geq 0\) and \(\dot{E}(t)<0\), \(E(t)\) is positive, zero, and decreasing \(E(t)\to 0\), as \(t\to\infty\). This implies that \(v(t)\to 0\), as \(t\to\infty\). To make this more abstract, we pick a non-negative scalar function \((v(x))\) that decreases along the trajectories of the agricultural drone, namely \(\dot{x}=f(x)\).  
\[\dot{v}(x)=\sum_{i=1}\frac{\partial v}{\partial x_{i}}\dot{x}_{i}=\sum_{i=1}^ {n}\frac{\partial v}{\partial x_{i}}f(x)=\
abla v(x).f(x) \tag{59}\]  
If \(\
abla v(x).f(x)<0\), \(\
abla v(x).f(x)>0\), and \(\
abla v(x),f(x)=0\), then the function is inwards, perpendicular, and away, respectively, with respective to the level sets of \(v(x)\). That means that the level sets are positive invariants. Therefore, stability in the sense of the Lyapunov law is summarized as follows.  
Let \(x^{*}=0\) be an equilibrium point of \(\dot{x}=f(x)\) and let \(\mathrm{v(x)}\) be a continuously differentiable function. If \(v(x)=0\) and \(v(x)>0\) for \(x\
eq 0\), and if \(v(x)\leq 0\), then the system \((x^{*})\) is stable at the equilibrium points. This implies the the system \((x^{*})\) with a fuzzy PID controller provides stability for agricultural drones.  
If \(\dot{v}(x)<0\), then the system \((x^{*})\) is asymptotically stable at the equilibrium points. If \(\dot{v}(x)\leq 0\) and \(\dot{v}(x)=0\), only at \(\mathrm{x}=x^{*}\), this implies that the system \((x^{*})\) is asymptotically stable at the equilibrium points. This indicates that the PID controller provides asymptotic stability for agricultural drones. In this instance, the Lyapunov stability proof offers a method to quantitatively prove the stability of a dynamical system for agricultural drones. Finding a function known as the \"Lyapunov function\" that becomes less valuable as the system deviates from its equilibrium state is the goal. This suggests that the agricultural drone system is stable because it tends to revert to equilibrium. Therefore, the proposed controllers were the most suited to stabilizing agricultural drones during their operation in agricultural fields.