Illustrative Examples
As stated in Chapter 3.2.3, the state variables of the ADMIRE benchmark aircraft are selected as AOA, angle of sideslip, roll rate, pitch rate, and yaw rate. The exogenous disturbance is chosen to be a constant vertical gust with a magnitude of 5m/s. The RC, LC, ROE, RIE, LIE, LOE, and rudder are the control inputs. The tracking selection matrix is chosen as S_{r} = diag {1, 1, 1} which means that AOA, sideslip angle, and roll rate are to be tracked. For all examples in this chapter, the command signal for AOA is set at 2 degree magnitude, 11 second duration pulse beginning at 1 sec. The reference signal of sideslip and roll rate is 0. The trimmed values of the ADMIRE aircraft equations are: M_{a} = 0.45, h = 3000m, V_{t} = 147.86m/s, a = 3.73743°, S = 0, T = 0.0752,5_{rc} = Sic = 0.0518 °, 6_{rie} = 0.036 18 °, S_{roe} = S_{tie} = 6i_{oe} = 0.03618°, S_{r} = 0.
The entire procedure of designing FTC which includes the fault models, redundancy analysis, and FTC synthesis is illustrated. The area of the control surface is used as the parameter, and the scheduling parameter is also a function of that as presented in Chapter 3.2.3.
For the 5th order aircraft system, the corresponding characteristic polynomial is .s'^{5} + «4a^{4} + a;js^{3} + a2s^{2}+a i s+«о. The relationship between the extent of damage to the inner elevons and the coefficients o_{4},03, a2, ai, ao is illustrated in Fig. 3.3. It can be seen that the coefficients 04,03,02,01,00 vary linearly with the degree of damage (proportional to the area of the inner elevons). Consequently, the maxima and minima on the area of inner elevons can be used as vertices in the polytope to describe the normal and faulty conditions.
Safety Control System. Design against Control Surface Impairments
FIGURE 3.3: Coefficients of characteristic polynomials.
The relationship between the area of the inner elevons and the developed fault models is shown in Fig. 3.4. With an increased degree of inner board elevon impairment (decrease of control surface area), the dominant eigenvalues of the fault models change from —3.8744 to —3.8088, which mean the deterioration in stability. The ADMIRE aircraft redundancy is analyzed based on Definitions 3.1 and 3.2 [27, 85]. Since the number of the required outputs is three and the number of independent control inputs is seven, the degree of redundant actuators is four, and the remaining control surfaces are sufficient to counteract the failures.
Several fault scenarios of inner elevon impairment have been considered, but due to the limited space, only the normal case and the worst case corresponding to 100% loss of the inner board elevons are presented in the response figures. In the state feedback example, the detailed performance of the nominal case, inner elevons loss of 50%, inner elevons loss of 75%, and inner elevons loss of 100% are collected in Table 3.1. In Table 3.2, the detailed indices are listed for the static output feedback example. However, in a practical system, the estimation of pis not always accurate. Therefore, a sensitivity analysis of the estimation error in p is also conducted. The analysis procedure is done by deliberately introducing ±20% error in estimation of p, and the performance indices are shown in Tables 3.3 and 3.4 corresponding to the state feedback and static output feedback cases, respectively.
For comparison purposes, a nominal controller that is based on robust control theory [89] and a reliable controller based on the idea in Ref. [21] have been designed. To illustrate the effectiveness of the proposed FTC design approach, the performance of the FTC is compared with that of the robust controller and
FIGURE 3.4: Relation between elevons damage degree and fault models.
the reliable controller through nonlinear simulations. In the case of the robust controller, the performance is not satisfied when the considered fault occurs. For the reliable controller, only a single controller is designed to deal with all the considered fault situations. The performance under both the normal and faulty cases is worse than that achieved from the proposed FTC. The nominal and fault models of the ADMIRE benchmark aircraft, the robust controllers, the reliable controllers, and the designed faulttolerant controllers under state feedback and static output feedback cases are given in Appendix D.
Example 1 (State Feedback Case)
Select the following weighting matrices when designing the robust controller, the reliable controller, and the proposed FTC based on LPV techniques:
The design technique for the robust controller is presented in Ref. [86] and the reliable controller is synthesized according to the idea in Ref. [21]. Nonlinear simulation responses of AOA and control surface deflections comparing the FTC and the robust controller are shown in Fig. 3.5. In the interest of space, only the RC deflection responses are displayed. Significant improvement in tracking performance with the FTC is achieved as shown in Figs. 3.5(a) and 3.5(b). The control surface deflections show that the healthy actuator is used to compensate for the effect of inner elevon impairments. Focusing on
TABLE 3.1: AOA performance indices comparison  state feedback.
Controller 
Performance 
Normal Case 
RIE 50% LIE 50% 
loss loss 
RIE 75% LIE 75% 
loss loss 
RIE 100% LIE 100% 
loss loss 
FTC 
Overshoot (%) 
0.71 
3.99 
5.68 
7.85 

Settling time (s) 
0.77 
0.74 
1.20 
1.33 

Robust 
Overshoot (%) 
0.03 
10.32 
21.85 
41.08 

controller 
Settling time (s) 
0.75 
1.10 
1.83 
4.32 

Reliable 
Overshoot (%) 
8.37 
9.07 
9.46 
9.86 

controller 
Settling time (s) 
1.38 
1.40 
1.40 
1.41 
the worst case in Fig. 3.5, the tracking performance of AOA is not acceptable for the robust controller. However, the designed FTC performs effectively in this situation. Comparison of the proposed FTC and the reliable controller is provided in Fig. 3.6. It can be seen that the proposed FTC has a superior performance to that of the reliable controller, where zero steady error with a very small transient is obtained. As indicated in Fig. 3.6(c), the remaining control surfaces are stepped up to counteract the negative effects caused by the impairments.
Table 3.1 indicates that the robust controller can still stabilize the system even for the worst case of inner elevon impairments. However, the corresponding performance has deteriorated considerably. The proposed FTC results in an overshoot index of 0.71% and a settling time index of 0.77 s which are not very far from those obtained from the robust controller under the normal case. By contrast, in the worst case, it is indicated that the performance indices corresponding to the proposed FTC with 7.85% overshoot and 1.33 s settling time are significantly better, as compared to those with 41.08% overshoot and 4.32 s settling time from the robust controller.
In the other two fault cases, the FTC has much better tracking performance than that of the robust controller. Moreover, the FTC performs significantly better than the reliable controller with less overshoot and shorter settling time in AOA response. In the worst case, the performances of overshoot and settling time achieved by the proposed FTC are improved by 20.39% (from 9.86% to 7.85%) and 5.67% (from 1.41s to 1.33s), respectively. ThereTABLE 3.2: AO A performance indices comparison  static output feedback.
Controller 
Performance 
Normal Case 
RIE 50% LIE 50% 
loss loss 
RIE loss 75% LIE loss 75% 
RIE loss 100% LIE loss 100% 
FTC 
Overshoot (%) 
7.11 
7.20 
7.48 
7.52 

Settling time (s) 
4.72 
4.75 
4.82 
5.07 

Robust 
Overshoot (%) 
0.42 
41.07 
Oscillatory 
Oscillatory 

controller 
Settling time (s) 
4.23 
4.32 
Oscillatory 
Oscillatory 

Reliable 
Overshoot (%) 
7.70 
7.76 
7.81 
8.08 

controller 
Settling time (s) 
4.89 
4.98 
5.02 
5.07 
fore, it is concluded that the proposed FTC is less conservative than the reliable controller.
Example 2 (Static Output Feedback Case)
In this illustrative example, the weight matrices when designing the robust, controller, the reliable controller, and the proposec FTC are selected to be:
ГЛ _ 0_{Gx7}
[o.2xD„J
Only the normal and the worst cases are provided in the response plots, and only RC and LC deflections are shown. The results of a comparison similar to that under the state feedback case are shown in Figs. 3.7 and 3.8, respectively. From the transient responses in Fig. 3.7(b), the FTC performs much better when the considered fault occurs. From the control surface deflection responses, the deflections under the fault case are a little larger than those under the nominal case since the remaining control surfaces are driven to counteract the fault effects. The compared results of the proposed FTC and the reliable controller are illustrated in Fig. 3.8. The comparison curves of the specified scenarios are illustrated to validate the effectiveness of the designed FTC.
FIGURE 3.5: Comparison results with FTC and robust controller via state feedback in nonlinear simulation.
The detailed performance indices are listed in Table 3.2. In the considered examples, the robust controller cannot stabilize the system when the more serious failures occur. During normal operation, the proposed method results in an overshoot of 7.11% and a settling time of 4.72 s, which are not very far
FIGURE 3.6: Comparison results with FTC and reliable controller via state feedback in nonlinear simulation.
from the indices obtained by the robust controller. Focus on the performances achieved by the reliable controller and the designed FTC. When compared with the reliable controller, the performance of overshoot with the proposed FTC is improved by 7.66%, 7.22%, 4.23%, and 6.93%. The performance of settling time with FTC is improved by 3.47%, 4.61%, 3.98%, and 0%, respectively. The comparison results confirm that the FTC corresponding to a certain degree of control surface impairments is less conservative than the reliable controller.
FIGURE 3.7: Comparison results with FTC and robust controller via SOF in nonlinear simulation.
FIGURE 3.8: Comparison results with FTC and reliable controller via SOF in nonlinear simulation.
Sensitivity Analysis
The scheduling parameter is a function of the control surface area. In a practical flight control system, the FTC is scheduled according to the estimate of the remaining control surface area. After the comparison among the proTABLE 3.3: Sensitivity analysisstate feedback.
Controller 
Performance 
—20% Error in P 
No Error 
+20% Error in P 
FTC with IE 
Overshoot (%) 
4.11 
3.99 
5.22 
50% loss 
Settling time (^{s}) 
0.76 
0.74 
1.12 
FTC with IE 
Overshoot (%) 
5.81 
5.68 
7.45 
75% loss 
Settling time (^{s}) 
1.21 
1.20 
1.29 
TABLE 3.4: Sensitivity analysisstatic output feedback.
Controller 
Performance 
—20% Error in P 
No Error 
+20% Error in P 
FTC with IE 
Overshoot (%) 
7.83 
7.20 
10.93 
50% loss 
Settling time (^{s}) 
4.93 
4.75 
5.31 
FTC with IE 
Overshoot (%) 
7.98 
7.48 
14.63 
75% loss 
Settling time (^{s}) 
5.22 
4.82 
6.95 
posed FTC, the robust controller, and the reliable controller, the sensitivity analysis of error in estimation of p is carried out. In Table 3.3, a ±20% error in estimation is considered for the state feedback case. Through sensitivity analysis of error in the estimated value of p. it is validated that the designed FTC can still stabilize the faulty system and achieve an acceptable level of performance when the given degree of estimation error occurs.
Similarly, sensitivity analysis under the static output feedback case is also conducted, and the corresponding indices are indicated in Table 3.4. From the analysis results, the proposed FTC still performs well when the estimation of p is not very accurate.
Summarizing the numerical examples of control surface impairment in both the state feedback and the static output feedback cases, it can be seen that the designed FTC can significantly improve the system performance in the event of fault cases, when compared with the robust and the reliable controllers. Furthermore, during nominal operation, the FTC achieves almost the same level of performance as the robust controller. From the sensitivity analysis, the proposed FTC performs satisfactorily in the case of error in estimation of p. Therefore, the effectiveness of the proposed approach has been validated by the extensive nonlinear aircraft simulations.
Conclusions
In this chapter, an FTC design procedure against control surface impairments is investigated. The control surface impairments are modeled as an LPV polytopic model using a parameter which is proportional to the effective control surface. The actuator redundancy of the system has been examined under various control surface impairments. Subsequently, FTC laws are designed using both the state and static output feedbacks. Nonlinear simulations have been carried out using the parameters from the ADMIRE aircraft, and the results have shown that the proposed techniques can indeed maintain the essential properties of the aircraft under various fault conditions. The performance of the FTC has been compared against that of robust controller and reliable controller. It concludes that the conservatism has been considerably reduced.
Notes
The main contribution of this chapter is to develop the entire procedure of designing an FTCS against control surface impairment failures. The procedure includes developing the aircraft failure models in which the control surface area is considered as a fault parameter, analyzing redundancy under the fault conditions, and designing the FTC based on LPV techniques through the optimization of LMIs to counteract control surface impairments. Compared with robust and reliable controller, the conservatism of the proposed controller has been considerably reduced.