# Adaptive Recursive Sliding Mode Control (ARSMC)-Based UAV Control for Future Smart Cities
## 4 Recursive Control Design
\[\vdots\qquad\vdots\qquad\vdots\qquad\vdots \tag{20}\]1  
The resultant error dynamics of \(e_{\xi_{n}}\) can be obtained as:  
\[\vdots\qquad\vdots\qquad\vdots\qquad\vdots \tag{20}\]2  
Stability Analysis of Recursive Backstepping: The error dynamics of the closed-loop system in coordinate \(e_{\xi}=\left[e_{\xi_{1}},e_{\xi_{2}},\ldots\ldots\ldots e_{\xi_{n}}\right]^{T}\) can be written as: \(\dot{e}_{\xi}=A_{e_{\xi}}e_{\xi}\)  
\[\left[ \begin{array}{c}\dot{e}_{\xi_{1}}\\ \dot{e}_{\xi_{2}}\\ \dot{e}_{\xi_{3}}\\ \vdots\\ \dot{e}_{\xi_{n}}\end{array} \right]=\left[ \begin{array}{ccccc}-k_{1}&1&0& \cdots&0\\ -1&-k_{2}&1&\ddots&0\\ 0&-1&-k_{3}&\ddots&0\\ \vdots&\ddots&\ddots&\ddots&1\\ 0&0&0&\ddots&-k_{n}\end{array} \right]\left[ \begin{array}{c}e_{\xi_{1}}\\ \dot{e}_{\xi_{2}}\\ e_{\xi_{3}}\\ \vdots\\ e_{\xi_{n}}\end{array} \right] \tag{40}\]  
The above matrix demonstrates that the off-diagonal elements possess skew-symmetric properties. To evaluate the stability of the transformed system, a Lyapunov candidate function can be utilized. Lyapunov functions are scalar functions used to establish the equilibrium stability of an ordinary differential equation (ODE). These functions are essential to stability and control theory and are named after the Russian mathematician [PERSON]. For several classes of ODEs, the presence of the Lyapunov function is a necessary and sufficient condition for stability. Although constructing a Lyapunov function for ODEs is not a general technique, it is known for many specific cases. In essence, a Lyapunov function is a function that has positive values everywhere except at the equilibrium and decreases (or remains non-increasing) along every trajectory of the ODE. The primary advantage of Lyapunov-function-based stability analysis of ODEs is that the exact solution of the ODE is not required, whether analytical or numerical [40]. The candidate function for the considered system is here:  
\[V=\frac{1}{2}e_{\xi_{1}}^{2}+\frac{1}{2}e_{\xi_{2}}^{2}+\ldots\frac{1}{2}e_{ \xi_{n}}^{2} \tag{41}\]  
\[\dot{V}=-\sum_{m=1}^{n}k_{m}e_{\xi_{m}}^{2} \tag{42}\]  
_Adaptive Recursive Sliding Mode Control Strategy_  
In summary, adaptive recursive sliding mode control (ARSMC) combines three different control techniques--adaptive control [39], recursive control [40], and sliding mode control [36]--to achieve robust and adaptive control of complex nonlinear systems with uncertain dynamics. The sliding mode control provides robustness to disturbances and uncertainties, while the adaptive control adjusts the controller parameters in real-timebased on estimation of the system states and model parameters. The recursive control updates the controller parameters based on the current state of the system, allowing it to track changes in the system dynamics and adapt to them in real-time. The ARSMC has been successfully applied to a wide range of complex nonlinear systems, including robotics, aerospace, and power systems. Its effectiveness and robustness have been demonstrated through simulation and experimental studies, making it a promising approach for advanced control of complex nonlinear systems. ARSMC is an advanced control strategy that combines adaptive control, recursive control, and sliding mode control (SMC) techniques to improve the control performance of complex nonlinear systems. In ARSMC, the system model is not required to be known exactly and the controller adapts to the changes in the system dynamics to achieve better control performance. The controller is designed recursively, which means that the current state of the system is used to update the controller parameters for the next control action. This allows the controller to track the changes in the system dynamics and adapt to them in real-time. The sliding mode control technique is used to force the system states to converge to a sliding surface that is defined based on the system model and the desired control objectives. The sliding mode provides robustness to the system against disturbances and uncertainties.