# Performance-Constraint Fault Tolerant Control to Aircraft in Presence of Actuator Deviation
## 5 Compared Results in a Prototype  
This section verifies the proposed yaw-vertical controller based on GTMAX prototype [36] simulation platform, meanwhile, controllers of classical FTC and asymptotic convergence based on quadratic Lyapunov function (QLAC) are also designed to compare the performance of dynamic response, rate constraint, actuator-deviation rejection. The controller parameters are given as \begin{tabular}{c c} \hline \hline \(BL\_PFTC\_FT\) & \(K_{1}=K_{2}=\text{diag}(10,20),\mu=0.8,k_{\tilde{h}}=2,\delta_{\text{max}}=0.28\) \\  & \(\gamma=2,\eta=10,\omega=10,k_{\omega}=0.2,k_{r}=0.3\) \\ FTC & \(K_{1}=K_{2}=\text{diag}(10,20),\mu=0.8\) \\ QLAC & \(K_{1}=K_{2}=\text{diag}(10,20)\) \\ \hline \hline \end{tabular}   
The nominal parameters of the prototype used in the controller design are \(I_{xx}=3.246\), \(I_{yy}=11.229,I_{zz}=9.856,N_{r}=-0.59\), \(N_{col}=-72.74\), \(N_{lr}=69.3\), \(Z_{w}=-0.61\), \(Z_{col}=-90.89\). The initial state is in hover, tail rotor speed settled as 5700 rpm, the roll and pitch channels have stabilized by a separate linear controller. Figure 2 compares the dynamic convergence of BL_PFTC_FT, FTC, and QLAC in the initial stage. Obviously, the proposed BL_PFTC_FT controller comparing with traditional FTC and QLAC has faster convergence speed and smaller transient error. Taking Figure 2a,b as examples, the yaw and vertical motion rates under the control of BL_PFTC_FT achieve stable within 0.2 s, while the FTC's convergence time reaches 0.4 s and 0.8 s respectively, and the maximum transient error exceeds 1.5 deg/s and 0.4 m/s. Furthermore, the fault-tolerant performance of the three methods is investigated in simulation experiments. As shown in Figure 3, total control perturbation caused by fluctuation of main rotor pitch mechanism. The vertical rate oscillation controlled by BL_PFTC_FT is confined in the range of \(-0.1\) m/s to \(0.1\) m/s as shown in Figure 4, while the oscillation of FTC and QLAC are all over \(0.2\) m/s. Similarly, the stable height of BL_PFTC_FT is closer to the target height of \(10\) m in Figure 5, and the steady-state error is smaller than FTC and QLAC. The fault-tolerant capability has been enhanced to accommodate the total distance aroused by var  
Figure 3: Actuator fluctuation of main rotor. Figure 2: Transient performance comparison of yaw and vertical motion. (**a**) Yaw rate; (**b**) Yaw angle; (**c**) Vertical velocity; (**d**) Height. The fault tolerance performance of yaw channel is also investigated separately in the simulation. The output fluctuation of tail rotor control \(\Delta\delta_{ped}\) is set to the same amplitude as of the main rotor \(\Delta\delta_{col}\). In Figure 6, it can be seen that the yaw rate oscillation of BL_PFTC_FT is always within in the range of \(\pm\)0.1 deg/s, which is smaller than FTC and QLAC methods. Figure 7 shows the yaw angle \(\psi\) changing process, obviously, the amplitude of oscillation controlled by FTC or QLAC is larger than BL_PFTC_FT. Therefore, the BL_PFTC_FT controller has improved the capability of helicopter's resistance to the output fluctuation of tail rotor pitch servo mechanism. Moreover, when yaw-vertical channels exist maneuvering fluctuations simultaneously, the simulation results verify the effect of auxiliary state \(\Psi\) designed in BL_PFTC_FT to improve the resistance to actuator fluctuations in Figure 8. The yaw angle and vertical rate are chattering augment without the auxiliary state as shown in Figure 8a,c, on the contrary in the diagrams of Figure 8b,d. Further, the fault tolerance performance in yaw motion control is considered suffering paddle speed fluctuation when the unmanned helicopter is cruising at a horizontal speed of 40 km/h. the oscillation amplitude of speed \(\Delta\omega_{tr}\) is set to be 500 r/min sinusoidal wave. Figure 9 illustrates the stabilizing process to yaw channel of the three controllers.