# A Robust Fixed-Time Sliding Mode Control for Quadrotor UAV
## 4 Numerical Simulation Results  
In this section, the results obtained are validated by numerical simulations and a comparison is made with the proposed NTSMC algorithm. This will allow us to determine the efficiency and superiority of the trajectory-tracking controller by visualizing the fast response speed, high tracking accuracy and minimization of perturbation; the following controller is compared concerning our proposed algorithm. To verify the robustness of the proposed control algorithm, the Dryden model has been implemented in the simulations in order to recreate a real environment with gusts of wind that could affect the trajectory of the aerial vehicle. The Dryden wind model has become a valuable tool in the development and testing of UAVs, as it allows engineers to study and better understand how wind affects vehicle behavior and stability in different situations. By creating scaled-down models of UAVs and testing them in wind tunnels, engineers can measure and analyze the aerodynamic forces acting on the vehicle in different wind conditions. These tests allow engineers to make improvements to the UAV design to maximize stability and maneuverability, reduce energy consumption and improve flight efficiency and safety. Overall, the use of the Dryden wind model has become an essential tool in creating and improving UAVs, enabling the development of more advanced and effective aerial vehicles for a wide variety of applications [33]. Non-singular terminal mode sliding control (NTSMC) is a highly accurate and robust control technique that has been successfully used in a wide range of applications. NTSMC combines the advantages of sliding mode sliding control (SMC) and terminal mode control (TMC) to achieve accurate and fast control of dynamic systems. Compared with conventional SMC, NTSMC has the advantage of avoiding the problem of the singularity phenomenon, which may occur when the system is close to its equilibrium point. In addition, NTSMC can guarantee the global convergence of the system to its equilibrium point, even when the system model is uncertain or nonlinear. The NTSMC was published in reference [34], where a nonlinear dynamical system is expressed as follows:  
\[ \begin{split}\dot{x}_{1}&=x_{2}\\ \dot{x}_{2}&=f(\mathbf{x})+g(\mathbf{x})+b(\mathbf{x})u\end{split}  \tag{43}\]  
where \(\mathbf{x}=\left[x_{1},x_{2}\right]^{\mathrm{T}}\) is the system state vector, \(f(\mathbf{x})\) is the smooth nonlinear function of \(\mathbf{x}\) and \(g(\mathbf{x})\) represents the uncertainties and perturbations satisfying \(\|g(\mathbf{x})\|\leqslant l_{g}\) where \(l_{g}>0\), and \(u\) is the scalar control input. The proposed NTSMC model is described as follows:  
\[s=x_{1}+\frac{1}{\beta}x_{2}^{p/q} \tag{44}\]  
where \(\beta>0\) is a design constant, and \(p\) and \(q\) are positive odd integers, which satisfy the following condition:  
\[p>q \tag{45}\]  
For system (43) with the NTSMC (44), the obtained control input is:  
\[u=-b^{-1}(\mathbf{x})\bigg{(}f(\mathbf{x})+\beta\frac{q}{p}x_{2}^{2-p/q}+\big{(}l_{g} +\eta\big{)}\operatorname{sgn}(s)\bigg{)} \tag{46}\]  
where \(1<p/q<2\), \(\eta>0\); stability proof is shown in reference [34]. Now, to represent the NTSMC control in a Quadrotor aircraft, the sliding variable sy is expressed as follows:  
\[s_{y}=e_{y}+\frac{1}{\beta_{y}}e_{y}^{py/q} \tag{47}\]  
Defining:  
\[e_{y}=y^{d}-y \tag{48}\]differentiating twice:  
\[\ddot{x}_{y}=\dot{y}^{d}-\ddot{y} \tag{49}\]  
and from Equation (5b) and considering that \(r\) converges to 0 after a certain time. \[\ddot{e}_{y}=\ddot{y}^{d}-g\phi+\Gamma_{y} \tag{50}\]  
Then, proposing \(\phi^{d_{y}}\) as the virtual input of the system (50), \(\ddot{x}_{y}\) converges to 0 based on [34].