## 6 Experimental Analysis
### Simulation Verification  
#### 6.1.1 Optimization Analysis of Controller Parameters  
The simulation takes the pitch angle as an example (the optimization process of other attitude angles is similar). The experiment uses PSO, FPSO and FOFPSO to optimize the parameters of the controller, and it compares them with the unoptimized controller. In the experiment, the expected input control signal of the controller \(r(t)=0.61\) rad; the experimental results are shown in Figure 7:  
For the unoptimized ADRC, there is a large overshoot and steady-state error; the controller optimized by PSO and FPSO has higher performance, but there is a certain shock in the rising period, resulting in an overshoot. After the controller is optimized by the FOFPSO algorithm, the response effect of the controller is better, and there is almost no overshoot and oscillation. As ADRC arranges the transition process of the input signal in the TD phase and also tracks the transition curve based on the idea of \"small error and large gain\", which directly affects the rise time of the controller response, it shows that the rise time of the four algorithms is basically the same, and the main difference lies in the overshoot and steady state value.  
Figure 8 shows the process of optimizing the parameters of the quadrotor vehicle controller by PSO, FPSO and FOFPSO algorithms, respectively; the three algorithms  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter Symbol** & **Physical Meaning** & **Value** & **Unit** \\ \hline \(m\) & Total mass of quadrotors & 1.4 & kg \\ \(d\) & Rotor arm length & 0.225 & m \\ \(J_{x}\) & \(X\)-axis moment of inertia & 0.0211 & kg\(\cdot\) m\({}^{2}\) \\ \(J_{y}\) & \(X\)-axis moment of inertia & 0.0219 & kg\(\cdot\) m\({}^{2}\) \\ \(J_{z}\) & \(X\)-axis moment of inertia & 0.0366 & kg\(\cdot\) m\({}^{2}\) \\ \(C_{t}\) & Propeller pull coefficient & \(1.022\,\times\,10^{-5}\) & N\(/(\text{rad}/\text{m})^{2}\) \\ \(C_{m}\) & Torque coefficient & \(1.401\,\times\,10^{-7}\) & N\(\cdot\) m\(/(\text{rad}/\text{m})^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model parameters of quadrotor aircraft.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Range of Value** & **Parameter** & **Range of Value** \\ \hline \(\beta_{01}\) & \(1\sim 500\) & \(c\) & \(1\sim 400\) \\ \(\beta_{02}\) & \(1\sim 4000\) & \(h_{1}\) & \(1\sim 400\) \\ \(\beta_{03}\) & \(1\sim 8000\) & \(b_{0}\) & \(1\sim 100\) \\ \(r\) & \(1\sim 400\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter range of ADRC.  
Figure 7: Step response curve of pitch angle.  
optimize the controller with same initial value. With the increase in the number of iterations, the particles continue to find the optimal value in the feasible space. Among them, the traditional PSO algorithm has poor optimization ability, and the early global exploration ability is not strong and cannot carry out large-scale exploration. Due to the existence of fuzzy logic, FPSO has strong optimization ability in the early stage, but it is easy to fall into the local optimal value in the later stage. Lastly, the FOFPSO algorithm proposed in this paper adds the characteristic of fractional order on the basis of FPSO, the global search ability of particles is stronger, and the optimization accuracy is higher than that of PSO and FPSO.  
The parameters of the optimized pitch angle controller are shown in Table 3: Optimized parameters of ADRC.