## 4 Numerical Simulations
 \tag{139}\]  
From (53), the matrix \(\mathbf{P}_{3x}\) is  
\[\mathbf{P}_{3x}= \begin{bmatrix}1.5796&0.0220\\ 0.0220&0.0046\end{bmatrix}  \tag{140}\]  
Assume \(M^{+}=2.5\) in AIBC; then, \(w=20>1.2\Big{(}\frac{max(d_{\theta})+}{J}+M^{+}\Big{)}=19.61\) is used to satisfy (106). The step-wise procedures to implement the control law are briefly summarized as follows:  
1. Given the desired position and altitude trajectory \(x_{d}\), \(z_{d}\), and their successive derivatives, \((\dot{x}_{d},\ddot{x}_{d},x_{d}^{(3)},x_{d}^{(4)})\) and \((\dot{z}_{d},\ddot{z}_{d},z_{d}^{(3)},z_{d}^{(4)})\);
2. Given the control gain pair \((k_{pz},k_{iz},k_{dz})\), then determine the control force \(F\) from (37);
3. Determine the control torque \(M\): 1. For CPID, calculate the desired attitude \(\theta_{d}\) from (57) and (A3). Then, the control torque is obtained from (A7). It should be noticed that the derivative of desired attitude, \(\dot{\theta}_{d}\) and \(\ddot{\theta}_{d}\), is obtained by using the APD to \(\theta_{d}\); 2. For IBC, given the control gain pair \((k_{px},k_{ix},k_{dx})\), solve the LMI (50) and then verify (51). Then, the control torque can be obtained from (54). It should be noticed that the derivative of virtual controls, \(\dot{\varphi}_{2}\) and \(\dot{\varphi}_{1}\), is obtained by using the APD from \(\varphi_{2}\) and \(\varphi_{1}\), respectively;For AIBC, given the control gain pair \((k_{p,x},k_{ix},k_{dx})\), solve the LMI (50) and then verify (51). Then, the control torque can be obtained from (94) so that the related quantities can be found in (95)-(105) accordingly;
4. Based on the calculated control force and control torque, compute the desired motor speed from (2);
5. Feed the desired speed to BLDC to generate the computed controls \((F,M)\). The simulation results are shown in Figures 3-11. The evolution of external disturbances is shown in Figure 3. Figure 3a,b depicted the time-varying disturbances and the state-dependent viscous drag, respectively. Figure 3a implies that the properties of external disturbances are \(\max(|f_{x}|)=0.2243\), \(\max(|f_{z}|)=0.2612\), \(\max(|f_{\theta}|)=0.049\), \(\max(|f_{x}|)=0.3321\), \(\max(|f_{x}|)=0.2938\), and \(\max(|f_{\theta}|)=0.056\). The zero-mean Gaussian white noise with variances of \(10^{-3}\) and \(10^{-5}\) are considered for the measurement noise of \((x,\hat{x},z,\hat{z})\) and \((\theta,\hat{\theta})\), respectively. The measured and actual states are illustrated in Figure 4. The actual states are collected from the integral of the system dynamics based on different control approaches. The measured states are the actual states that add the measurement noise, which will be used in the feedback control process. For the CPID, the APD is used for estimating the desired angular velocity \(\hat{\theta}_{d}\) and the desired angular acceleration \(\hat{\theta}_{d}\) from desired attitude \(\theta_{d}\) defined in (A3). For IBC, the APD is utilized to estimate the time derivative of virtual controls, that is, estimate \(\phi_{1}\) and \(\phi_{2}\) from \(\varphi_{1}\) and \(\varphi_{2}\) by the APD. Both APDs with a cutoff frequency of 10 Hz are considered. On the contrary, regarding the proposed AIBC, it does not need to use the APD since the time derivatives of the virtual controls are acquired by the analytic approach. The comparison of tracking performance for different approaches is revealed in Figures 5 and 6, respectively. The corresponding root-mean-square error (RMSE) of the resulting tracking errors is summarized in Table 3. The results reveal that the proposed AIBC has the best tracking performance compared with the CPID and IBC. In fact, due to the numerical differentiations, the rotor speeds of CPID and IBC are saturated, which may induce flight instability. Figure 4: Comparison of measured and actual states for different approaches. (**a**) CPID; (**b**) IBC; (**c**) AIBC. Figure 5: Comparison of the desired command and system response in the \(x\)-direction for different approaches.