# Simulation Studies

## Simulation Environment Description

The trimming condition of the Boeing 747 transport aircraft is indicated in Table 7.1, while the control authority of the RE and LE is presented in Table 7.2. The sampling interval for the Boeing aircraft control is chosen as 0.01s.

The reference signal is generated after passing through the following pre- filter:

where the original command в* is set with the magnitude of 5.74° and the duration of 15 s.

## Simulation Scenarios

The simulation scenarios are briefly summarized in Table 7.3. The LE outage (^2=0) is invoked at 13s of the simulation. The amount of time consumed by the FDD unit is assumed to be Is such that reliable FDD results can be achieved for control reconfiguration. The sensor gyro for measuring the pitch rate is available in Case I. while the sensorless pitch rate case is examined in

TABLE 7.1: Trimming values of the Benchmark Aircraft.

 Symbol Physical meaning Value Altitude at the trimming condition 7000 m Airspeed at the trimming condition 230 m/s AOA at the trimming condition 0.9283° Pitch angle at the trimming condition 0.9283° Pitch rate at the trimming condition 0 Engine thrust at the trimming condition 41631N RE deflection at the trimming condition 0 LE deflection at the trimming condition 0 Horizontal stabilizer deflection at the trimming condition 0.7334°

TABLE 7.2: The operating limits of the elevators.

 Control surface Symbol Amplitude limit Rate limit Right elevator Sre [—20°,20°] [-37%,37°/s] Left elevator he [—20°,20°] [—37%,37%]

TABLE 7.3: The simulation scenarios.

 Case I-A Case I-B Case II-А Case П-В The LE is outage V У У V q can be measured by the gyro V У q is reconstructed by SMO У V FDD time is 1 s V У V V Model uncertainty V У V V FDD inaccuracy у У V у Noises in measurements У у Input delay v/ У

Case II. Further, in terms of the scenarios set by Refs. [171, 191, 193], the phenomena including model uncertainties, noise measurements, input delay, and inaccurate estimation of actuator effectiveness factors are also introduced in the two scenarios. There exists 20% mismatch in the mass of the aircraft, and also in the aerodynamic coefficients (Cl and Cm). The noise with a mean of 0 and covariance of 0.01 is injected into each measurement channel. The input delay is fixed 0.01 s as well as the sampling interval. The estimation of the RE effectiveness indicator is 90% of its true value, while the estimated value of the LE effectiveness indicator is 50% of its true value.

Three types of fault accommodation architectures are compared in the simulations. They are: (1) the SMC-based fault accommodation without accounting for actuator physical constraints; (2) the adaptive SMC-based fault with consideration of actuator amplitude bounds; and (3) the developed accommodation scheme with consideration of both actuator amplitude and rate limits. In the interest of brevity, the three schemes are named as (1) FTFC, (2) FTFC-AL, and (3) FTFC-ARL, respectively. The following metric is defined to quantitatively compare the selected schemes,

where [f j ,f2] covers the simulation duration. The index in Eq. (7.81) describes the performance over the simulation interval considering both tracking error and actuation movements.

## Results of Case I and Assessment

(1) Case I-A

As depicted in Fig. 7.4, in the event of LE outage, the pitch angle response becomes oscillatory when the reference input tends to the trimming value (0.9283°) under the FTFC scheme. In contrast, the FTFC-AL and FTFC- ARL can preserve an acceptable level of pitch tracking performance. Fig.

7.5 reveals the resulting deflections of the elevators. The RE governed by the FTFC reaches the magnitude limits (20°) during the transient process. However, the RE can stay within the amplitude bounds to compensate for the impact due to the LE damage, when the FTFC-AL and FTFC-ARL are commissioned. Key observations of Fig. 7.6 include that: (1) the RE deflection rate saturation is encountered in the presence of both the FTFC and FTFC- AL schemes; and (2) the RE deflection rate under the FTFC-ARL is within the allowable range ([—37°/s, 37°/s]). It should be mentioned that the pitch rate of the FTFC-ARL is below those of the FTFC and FTFC-AL. This response coincides with the fact that the RE reacts less aggressively to avoid the control authority exhaustion.

The quantitative performance indices are included in Table 7.4. The peak values of the RE deflection are 20°, 10.22°, and 6.47°, corresponding to the FTFC, FTFC-AL, and FTFC-ARL, respectively. The RE rate exhaustion (37 °/s) is present under the FTFC and FTFC-AL systems. On the other hand, the peak value of the RE rate is 23.87 °/s in the case of FTFC-ARL. The FTFC-ARL also outperforms the FTFC and FTFC-AL in terms of the index eperf. The enhanced rate from the FTFC to the FTFC-ARL is 85.78%

FIGURE 7.4: The responses of aircraft states in Case I-A.

FIGURE 7.5: The deflections of elevators in Case I-A.

FIGURE 7.6: The deflection rates of elevators in Case I-A.

• (from 44.79 to 6.37), while the improved percentage from the FTFC-AL to the FTFC-ARL is 40.36% (from 10.68 to 6.37).
• (2) Case I-B

Figs. 7.7-7.9 illustrate that the FTFC-ARL scheme can achieve superior performance than those under the FTFC and FTFC-AL schemes, when the measurement noises and input delay are taken into account. To be more specific, the indices listed in Table 7.4 confirm that the LE deflection and deflecTABLE 7.4: Performance comparison in Case I.

 FTFC FTFC-AL FTFC-ARL Case I-A 20 10.22 6.47 37 37 23.87 44.79 10.68 6.37 Case I-B 20 10.85 6.61 37 37 35.14 60.36 23.86 18.97

tion rate are limited to the allowable ranges when the FTFC-ARL is engaged. The defined index achieved by the FTFC-ARL is better than those under the FTFC and FTFC-AL. Hence, in spite of more realistic factors, the presented FTFC-ARL system can achieve superior performance than that under the FTFC and FTFC-AL schemes.

## Results of Case II and Assessment

(1) Case II-A

It is highlighted in Fig. 7.10 that the FTFC-AL and FTFC-ARL ensure that the pitch angle can track the reference input after the LE failure. Instead, the FTFC scheme results in the pitch angle excursions. The RE managed by the FTFC-AL and FTFC-ARL is operating within the travel limits, as

FIGURE 7.7: The responses of aircraft states in Case I-B.

FIGURE 7.8: The deflections of elevators in Case I-B.

FIGURE 7.9: The deflection rates of elevators in Case I-B.

evidenced in Fig. 7.11. The RE deflection of the FTFC-ARL is smoother in comparison with those of the FTFC-AL and FTFC. What is interesting to see further is that the pitch angle tracking of the FTFC-ARL is slightly slower than the tracking of the FTFC-AL. thereby avoiding the exhaustive usage of RE. Fig. 7.12 exhibits the attractive benefit of the FTFC-ARL. not exceeding the actuator rate limits, in addition to keeping the flight safety. In this sense, the performance of the FTFC and FTFC-AL is inferior to that of

FIGURE 7.10: The responses of aircraft states in Case II-A.

FIGURE 7.11: The deflections of elevators in Case II-A.

the proposed FTFC-ARL. Finally, as can be seen in Fig. 7.13, the proposed SMO is capable of recovering the pitch rate information, which is reliable for the FTFC purpose.

Table 7.5 indicates that the RE with the peak deflections of 10.24° and 6.66° is managed to counteract the LE outage under the FTFC-AL and FTFC-ARL, respectively. The RE actuation rates continuously touch the physical bounds in the event of the FTFC and FTFC-AL. However, the peak deflection rate of the RE is 24.31 °/s under the control of the developed FTFC-

FIGURE 7.12: The deflection rates of elevators in Case II-A.

ARL. The defined index ерег/ is significantly improved by 85.19% (from 44.90 to 6.65). The FTFC-ARL also attains the superior performance compared to the FTFC-AL, with 38.48% improvement of the defined metric (from 10.81 to 6.65).

(2) Case II-B

In addition to Case II-A, the noise is injected into each measurement channel. while the input delay is fixed as the sampling interval. Figs. 7.14-7.17 show that the FTFC-ARL scheme can guarantee the flight safety in the event of LE outage and unreliable gyroscope. The FTFC-ARL can also ensure that the RE works within the physical limits during the fault accommodation process. The indices in Table V exemplify that the developed FTFC scheme outperforms the FTFC and FTFC-AL schemes in the realistic simulation environment. It should be noticed that the performance of Case II-B achieved by the designed

TABLE 7.5: Performance comparison in Case II.

 FTFC FTFC-AL FTFC-ARL Case 11-A 20 10.24 6.66 37 37 24.31 44.90 10.81 6.65 Case II-B 20 11.05 6.72 37 37 35.26 60.56 26.67 19.34

FIGURE 7.13: The actual (solid line) and reconstructed (dashed line) pitch angle (top left) and pitch rate (top right) in the proposed scheme, and the estimated errors of pitch angle (bottom left) and pitch rate (bottom right).

FIGURE 7.14: The responses of aircraft states in Case II-B.

scheme is slightly degraded than that of Case II-A, since the measurement noises and input delay are involved.

Finally, it is worth emphasizing that although the Boeing 747 transport aircraft in longitudinal motion has been selected to demonstrate the concept

FIGURE 7.15: The deflections of elevators in Case II-B.

FIGURE 7.16: The deflection rates of elevators in Case II-B.

and effectiveness, the proposed design strategies are general and applicable for different types of aircraft.

FIGURE 7.17: The actual (solid line) and reconstructed (dashed line) pitch angle (top left) and pitch rate (top right) in the proposed scheme, and the estimated errors of pitch angle (bottom left) and pitch rate (bottom right).

# Conclusions

Unexpected magnitude and rate limiting in healthy flight actuators can reduce the stability margin, and even deteriorate the safety of the post-failure aircraft. A new safety flight control scheme against actuator failures is proposed by incorporating both the actuator amplitude and rate bounds. Further, potential sensor fault on the measured angular rate is considered in the design procedure. In the absence of a reliable gyro onboard, the sensed pitch rate is delivered into the safety control loop. In the presence of gyro sensor fault, which leads to the sensed angular rate becoming no longer accessible, the pitch rate information is reconstructed by a SMO for the safety control feedback.

# Notes

Compared to the conventional SMC, an advantage of the proposed sliding surface with integral action is that the system trajectory always starts from the sliding surface. Through the numerical simulations, we can find that the unique advantages of the proposed methodologies include that: under actuator failures, (1) the magnitude and rate saturation in healthy actuators can be notably prevented; and (2) the flight safety and tracking performance can be maintained whether or not the measured angular velocity is offered.

Despite that the proposed strategies are capable of tolerating flight actuator and sensor gyro failures, issues including modeling and sensing uncertainties have not yet been considered in the design. Investigation of these factors which may affect the performance of the overall safety control system is one of our future works.

Appendix A