## 5 Simulation and Analysis  
In order to prove the effectiveness of the proposed active fault-tolerant control method for the attitude system of air-ground platform with time-varying delay fault, the relevant system parameters and fault simulation conditions are designed as follows:  
\[A=\left[ \begin{array}{ccc}0_{3*3}&I_{3*3}\\ 0_{3*3}&0_{3*3}\end{array} \right],A_{d}=\left[ \begin{array}{ccc}0_{3*3}&0.1*I_ {3*3}\\ 0_{3*3}&0_{3*3}\end{array} \right],C=\left[ \begin{array}{ccc}I_{3*3}&0_{3*3} \end{array} \right],D=I_{6*6},\]  
\[J=\left[ \begin{array}{ccc}7.278\times 10^{-2}&0&0\\ 0&7.278\times 10^{-2}&0\\ 0&0&1.367\times 10^{-2}\end{array} \right],B=\left[ \begin{array}{cccc}0_{3*1}&0_{3*1 }&0_{3*1}&0_{3*1}\\ 0&-0.325&0&0.325\\ -0.325&0&0.325&0\\ 0.0203&-0.0203&0.0203&-0.0203\end{array} \right],\]  
Model parameters selection are based on the designed air-ground platform model in Figure 1. The external disturbance is assumed to be \(\left[-0.005\sin t\quad 0.005\sin t\quad-0.005\sin t\right]^{T}\), The time-varying delay terms are \(d(t)=0.1+0.2\cos t\) and \(F(t)=0.01\sin t\). When there is no fault, the attitude system of air-ground platform adopts PID control method in flight mode. The control rate is designed as:  
\[u_{c}=B^{+}(-k_{p}J\big{[}x_{1}\quad x_{2}\quad x_{3}\big{]}^{T}+k_{i}J\big{[} x_{4}\quad x_{5}\quad x_{6}\big{]}^{T}+k_{d}J\big{[}x_{4}\quad x_{5}\quad x_{6} \big{]}^{T}) \tag{41}\]  
where \(B^{+}\) is a pseudo inverse matrix, which satisfies \(B^{+}=B^{T}(BB^{T})^{-1}\). When the fault occurs, the system detects and estimates the fault and switches to fault-tolerant control mode. According to the fault analysis of Section 2, we assume that three rotors have different degrees of fault damage at the same time, which seldom happens in reality. The fault design is shown in Table 1. It is assumed that at 25 s, rotor 1, 2, and 3 have actuator bias fault and LOE fault, respectively, where the bias fault is expressed as: \(f_{2}=-0.03+0.01\sin t\), \(f_{3}=0.06+0.01\sin t\). We use MATLAB for simulation verification. The system simulation time is set to 100 s and the sampling time is 0.02. It is reasonable for rotor motor to respond to the attitude change in time in real situations. The initial value of attitude angle is \( \begin{bmatrix}0.1&5&160\end{bmatrix} ^{T}\) in degree, and the initial angular velocity is \( \begin{bmatrix}0.006&0.005&0.003\end{bmatrix} ^{T}\) in rad/s. The desired attitude angle is expressed in quaternion \( \begin{bmatrix}0&0&0&1\end{bmatrix} ^{T}\) and the desired angular velocity is \( \begin{bmatrix}0&0&0\end{bmatrix} ^{T}\) in rad/s. The fault detection threshold is set to 0.17, and the saturation constraint of the actuator mechanism is \(u_{max}=10N\). The simulation results are shown in Figures 3-6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fault Type** & **Rotor 1** & **Rotor 2** & **Rotor 3** & **Rotor 4** \\ \hline LOE & 0.5 & 0.2 & Normal & Normal \\ \hline Bias Fault & None & \(f_{2}\) & \(f_{3}\) & None \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fault assumption of air-ground platform in flight mode. Figure 4: **(a)** Description of fault estimation; and (**b**) description of fault estimation error. Figure 3: Comparison between fault-tolerant control effect and PID control effect. (**a**) Description of fault-tolerant control effect; and (**b**) description of PID control effect. Figure 3a shows that after the fault occurs, the active fault-tolerant control scheme designed based on time-varying delay system in the paper can effectively suppress the adverse effects caused by the fault. In contrast, PID control cannot achieve stable attitude control in Figure 3b. Figure 4a shows the system fault estimation error. When time-varying fault is introduced, the estimation error in Figure 4b increases significantly and then tends to zero, which shows the effectiveness of the proposed active fault-tolerant control method. Figure 5a shows the output of the system under PID mode and fault-tolerant control.