Chinese Journal of Aeronautics, (2015), 28(6): 1658-1666

JOURNAL OF

AERONAUTICS

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cn www.sciencedirect.com

Active fault-tolerant control strategy of large civil aircraft under elevator failures

Wang Xingjian, Wang Shaoping, Yang Zhongwei, Zhang Chao *

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

Received 11 March 2015; revised 19 June 2015; accepted 6 September 2015 Available online 30 October 2015

KEYWORDS

Active fault-tolerant control; Longitudinal control; Model following; Performance improvement coefficient;

Trimmable horizontal stabilizer

Abstract Aircraft longitudinal control is the most important actuation system and its failures would lead to catastrophic accident of aircraft. This paper proposes an active fault-tolerant control (AFTC) strategy for civil aircraft with different numbers of faulty elevators. In order to improve the fault-tolerant flight control system performance and effective utilization of the control surface, trimmable horizontal stabilizer (THS) is considered to generate the extra pitch moment. A suitable switching mechanism with performance improvement coefficient is proposed to determine when it is worthwhile to utilize THS. Furthermore, AFTC strategy is detailed by using model following technique and the proposed THS switching mechanism. The basic fault-tolerant controller is designed to guarantee longitudinal control system stability and acceptable performance degradation under partial elevators failure. The proposed AFTC is applied to Boeing 747-200 numerical model and simulation results validate the effectiveness of the proposed AFTC approach. © 2015 The Authors. Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Safety is the most important requirement for civil aircraft. In order to meet the increasing safety demand and flight performance,1 fault-tolerant control (FTC) technique has been widely used in flight control system to guarantee system security and reliability when malfunctions of actuators appeared.

* Corresponding author. Tel.: +86 10 82338917. E-mail addresses: wangxj@buaa.edu.cn (X. Wang), shaoping-wang@vip.sina.com (S. Wang), yzw_buaa@qq.com (Z. Yang), czhangstar@gmail.com (C. Zhang).

Peer review under responsibility of Editorial Committee of CJA.

Over the past decades, many researchers devoted their efforts to FTC and a large number of results have been obtained.

Generally, fault-tolerant control systems (FTCSs) are control systems which are able to accommodate the component failures automatically by maintaining overall system stability and acceptable performance.1 There are two types of FTCS, i.e., passive and active.2-5 Passive FTCS needs neither fault detection and diagnosis (FDD) nor controller reconfiguration, but they often have limited fault-tolerant capabilities when implemented on the aircraft to tolerate kinds of failures. In contrast to passive FTCS, active FTCS reacts to system component failures actively by reconfiguring control actions to acquire the stability and acceptable performance of entire system.

Active fault-tolerant control (AFTC) can be classified into many approaches based on control algorithms.6-8 Multimodel technique-based FTC can effectively estimate and

http://dx.doi.org/10.1016/j.cja.2015.10.001

1000-9361 © 2015 The Authors. Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

accommodate faults without fault detection and isolation (FDI). When some faults appear, multi-model technique-based FTC can automatically switch to the suitable reference model which is pre-designed to match a certain fault. However, the transient performance during mode switching needs to be further considered. Similar to multi-model-based method,9-11 model predictive control is another effective AFTC approach. Thanks to the ability of handling most of the challenges of FTC in a generic and systematic way, model predictive control can take many important factors into consideration such as input and state constraint, uncertainty and time delay. However, real-time implementation problem is still a challenge due to its complex and vast calculation.12 On other hand, it is not easy to determine controller parameters at the design stage of AFTC because of some uncertain factors such as the model error, unknown actuator faults and external disturbance. Fortunately, adaptive control provides an effective way to make the compensation for the stuck fault, loss of effectiveness and outage fault of actuators even though that information is not available.13-15 However, the estimation and adjustment of parameters need to be carried out step by step on-line in real-time, and sometimes the transient state dynamics of system cannot be guaranteed within a satisfactory level.16'17 Besides, much work has been devoted to new intelligent neural network techniques, which are employed to adap-tively compensate for the unknown time-delayed nonlinear effects and changes in model dynamics due to the faults. Despite some attractive advantages of those intelligent methods, much more efforts are still needed to improve the learning speed of neural network to meet the need of the real-time control system in practice.18-20 Other methods for FTC include linear parameter varying, pseudo-inverse and linear matrix inequality, etc. Among these methods,21 model following is an attractive candidate for FTCS, because its goal is to emulate the performance characteristics of a desired model, with or without failures. Combined with other methods,22-25 model following was widely used on different FTCS due to its simple control structure and easy implementation.

On the other hand, compared with lateral/directional control of the civil aircraft, there are less control surfaces for longitudinal motion control, just the elevators and trimmable horizontal stabilizer (THS). In order to achieve better performance and higher reliability, an effective FTC strategy is required for the longitudinal control.26 Ganguli et al.26 presented the design of a reconfigurable linear parameter varying (LPV) controller for Boeing 747-100/200 longitudinal axis where two elevator fault scenarios were contemplated. Simulation results with elevator fault showed that reconfigurable controller stabilized the faulted system at the expense of a factor of a designed one-third reduction in the tracking responsiveness.27,28 Some FTC techniques based on adaptive or sliding model techniques were also studied for longitudinal motion control of aircraft. However, those researches seldom took a good advantage of the capacity of available surfaces according to flight mission. For some flight missions, the available elevators are sufficient for aircraft to get an acceptable performance even without the assistance of THS. But if elevator faults are more serious and bigger pitch moment is required, flight performance improvement due to THS extra moment will be obvious. Based on the above analysis, it is known that THS can be utilized to provide extra pitching moment when most elevators fail and large-scale pitch movement is required.

In order to improve the fault-tolerant flight control system performance and effective utilization of the control surface, it is necessary to give a criterion to decide when and how to control THS. For this purpose, a performance improvement coefficient is proposed to decide when the THS should be used to generate the extra pitch moment. Then a suitable switching mechanism is also designed to determine when it is worthwhile to control THS to achieve acceptable performance. Furthermore, AFTC strategy with model following technique and THS switching mechanism is proposed for large civil aircraft longitudinal control to achieve better fault-tolerant control performance. Firstly, a baseline controller is designed for the normal system as the reference model, then a model following fault-tolerant controller is designed using the available elevators. THS is fully considered in the proposed AFTC strategy to improve system performance. Simulation results verify the effectiveness of the proposed fault-tolerant strategy.

2. System models and problem formulation

The longitudinal kinematic behavior of an aircraft can be described in a state-variable form as29

x = Ax + BU + d

.y = Cx where

1 - Zw

Xde Zde Mde

1 - Zw

u0 + Zq

-g0cos00 gp sin 60 1- Zw

(U0 + Zq)Mw -Mwg0sin60

C = [0 0 0 1 ]

x =[ u w q 6 ]T denotes the longitudinal states, with u the axial velocity, w the longitudinal velocity, q the pitch rate and 6 the pitch angle; de is longitudinal control input (i.e., elevator angle command); y = 6 is longitudinal output angle; d is the lumped effect of uncertainties and disturbances; A is the system matrix, B the control matrix and C the output matrix.30 The detail meaning of these variables in A and B the can be found in NASA report CR-2144.

From Eq. (1), we can find that there is only one control input Se for the longitudinal control in this traditional system model, that means all elevators are assumed to achieve the same output angle and they are considered as an unity. Actually, for a practical civil aircraft, there are more than one elevator surfaces and they can achieve different output angles. Based on this consideration, in this paper, we assume that there are 4 independent elevators and one THS in longitudinal control. Generally, the THS is not used for pitch control directly in the normal mode. When there are more severe mal-

functions of the elevators during the flight, if needed, THS can provide extra pitch moment for the longitudinal control just like the elevators to get a better control performance than the condition without using THS. When and how to use the available elevators and THS (alternative) to reconfigure the control law is what we will study in this paper.

Firstly, we make the following assumptions: Assumption 1. There are two or three faulty elevators and the faulty elevators cannot provide any pitch moment for the aircraft, but the remained elevators are in good condition.

Assumption 2. THS is available and controllable for fault-tolerant control.

By taking the faulty elevator and the THS into consideration, the aircraft longitudinal motion can be described in another different form Eq. (1) as

X1 = Ax1 + B1u1 + d1 У1 = Cx1

"X¿ e X1¿ e

B = Z¿ e ! B1 = Z1¿e Z¿ s

M¿ e M1¿e M¿ s

0 0 0

where x1 = x, y1 = y and d1 = d; the system matrix A and output matrix C are the same as those in Eq. (1), while the control matrix B1 changes as

and the input vector щ — [d1e ds ] contains available elevator (d1e) and THS (ds). The first column of B1 is the stability derivatives of the remained elevator and the second column is that of THS.

Generally speaking, the dynamic performance of elevator actuator is much better than the flight control system dynamic performance, which means the elevator deflection bounded in the limit of amplitude can be considered equal to the command generalized by the controller without time delay, or the delay time can be neglected. However, it is not reasonable to apply this concept to THS, mainly because THS is originally designed for longitudinal trimming. Due to large inertia, the rotate speed of THS is usually bounded by a low limitation. For example, the rotate speed of THS on B747 is strictly kept below 0.2 (0)/s when the aircraft is at high speed. THS can provide bigger pitch moment, but it will take more time to achieve the desired deflection than an elevator, and this problem will be taken into account in this paper.

The THS actuator deflection is assumed to be constrained

dmin 6 ds 6 ¿max (8)

where dmin and dmax are the lower and upper saturation limits;

similarly, the rotate speed is kept within limits

where dmin and dmax are the lower and upper rate limits.

The command for actuator is generalized in discrete-time domain by the flight control system with a sampling period of T. Considering the rate limit, we can take the rate of change as an approximate first-order difference in equation:

¿ 6 ¿s(t + T) -¿s(t) 6 ¿ ¿min 6 лт-т 6 ¿max

Combining Eqs. (8) and (10), the saturation limit of next sampling time can be defined as following two constraints:

n (t + T) = max(¿m x(t + T) = min(¿m

; ¿s(t) ; ¿s(t)

T¿ m T¿ m

The dynamics of the THS should have been taken into account, but for simplicity, here we assume that THS can deflect at its upper or lower speed immediately until it arrives at the command position given by the control system. Therefore, THS deflection of the next interval can be presented as

¿s(t + T) =

¿s(t) ¿s(t)

T¿ m T¿ m

¿s (t) 6 ¿ cmd ¿ s (t) > ¿ cmd

where d cmd is the command of THS deflection and d s(t + T) still subjects to the two constraints in Eq. (12).

3. Active switching fault-tolerant control strategy

Firstly, a baseline controller is designed for failure-free aircraft, so that the pitch angle of the aircraft can follow the flight command without steady-state error:

lime(t) = lim(r(t) - h(t)) = 0 (13)

tH t!1

where r(t) is the desired pitch angle and e(t) the error. Normal system with baseline controller will serve as a reference model. On this basis, a reconfigurable fault-tolerant controller will be designed to compensate the faulty elevators using model following method, which aims to recover the original system performance or to accept some degree of performance degradation. The control strategy is shown in Fig. 1. The baseline controller dominates the system under normal condition; when the elevator failures are detected by FDI, system will be switched to the reconfigurable controller.

Nowadays, most of the researches on FDI and FTC are carried out as a two separate entity.1 More specifically, most of the FDI techniques are developed as a fault diagnostic or state monitoring tool, rather than an integral part of FTCS. Accurate FDI techniques have been widely applied on modern large aircraft and these results can be utilized directly.

3.1. Reference model design

When all the elevators are under good condition, for plant Eq. (1), the controller dynamics is set to be

xc = Acxc + Bc(r - h) (14)

Controller switch

Baseline controller Actuators

Fault-tolerant controller —

Aircraft

Fig. 1 Structure of overall longitudinal control strategy.

¿ min 6 ¿ s 6 ¿max

where xc e R1 is the controller state, Ac e R1 and Bc e R1.

By combining Eqs. (1) and (14), a new state-variable form is constructed as

x A 0 x B d

— + u +

xc -BcC Ac xc 0 r

y = 6 = [C 0 ]

Defining the new system matrix Ag e R x and control

matrix Bg e R5x 1 as

A0 -BcC Ac

The controllability of augmented system is checked by rank (C0) = 5 where C0 is the controllable matrix:

C0 = iB„

A2B„ A;B„ a4b,

The augmented system Eq. (15) is controllable. Therefore, a control law can be designed as

u — Kxx + Kcxc

where Kx e R1x4 and Kc e R1x1, making all closed-loop poles of system have negative real part, which means the system is stable. With the control law Eq. (20), the closed-loop system can be written as

(21) (22)

x A + BKx BKc x 1 d

xc -Bc C Ac xc + r

y = 6 — [C 0 ]

Assuming that both disturbance and command are step input, d = d0 1(t) and r = r0 1(t), with d0 and r0 the amplitudes of step signals, the Laplace transform of system Eq. (21) is

Using the inverse Laplace transformation, the steady value of Eq. (23) is

" x(s) ' si -(A + BKx) - BKc -1 d0 1

,xc(s). BcC si - Ac .r<>. s

x(s) x(s)

lim — lims

t!1 _xc(s)_ s!0 _xc(s)_

si -(A + BKX) -BKc" -1 d0

BcC si - Ac r0

From Eq. (24) we conclude that the steady value of xc(t) is a constant, that is to say, lim xc(t) = 0; when Ac is set to be zero,

Eq. (14) can be written as

lim (r(t)-6(t))= 0 (25)

and the control law Eq. (20) can also be equivalent to u = Kxx + Kc J (r - 6)dt (26)

With the control law Eq. (26), the system can make a perfect tracking of the command with no steady-error; by choos-

ing the reasonable feedback gain Kx and Kc, the system can get a better dynamic performance.

Linear quadratic regulator (LQR) technique is applied in this paper to calculate the feedback gain Kx and Kc. The control law can be conveniently found by selecting the state weighting matrix Q and input weighting matrix R.

3.2. Model following fault-tolerant controller

When one or more elevators fail, fault-tolerant controller should be designed to utilize the available elevators and even the THS if necessary, aiming to recover the original system (close-loop with baseline controller) performance or to accept some degree of performance degradation. A model following method will be applied to this study, as shown in Fig. 2. The reference model can be expressed in the form as

xm Am xm + Bmum ym Cmxm

where we choose the closed-loop system with baseline controller as the reference model, which is derived from Eq. (15):

A + BKx BKc

-BcC A, 04x1 1

Cm — [ 0 0 0 1 0 ]

where xm — [ x xc ]

x T, u xc , um

r, y„

The plant to be controlled is the aircraft with elevators failure which is formally stated by Eq. (6). By introducing a new controller state as that in baseline controller design in Eq. (15), we get a new form of the plant with elevator failures:

xp — Apxp + Bp «p + Brr

yp — Cpxp

where xp — [x xc ]T, «p — [d1e ds ]T; yp = 6 is the output. The relative matrixes in Eq. (29) are A 01

Ap —

Bp —

Br —

04x1 1

iCp — [0 0 0 1 0 ]

The error between the state of reference model Eq. (27) and plant Eq. (29) is defined as

e xm xp (31)

Fig. 2 Diagram of model following method.

A control law to calculate the input up is defined in the form

Hp = Kee + KmXm

where Ke e R2x4 and Km e R2x4. Then the differential of error can be written as

= (Ap - BpKe)e + (Am - Ap - BpKm)Xm

when the gain Km satisfies

Am — AP — BPKm = 0

we have

e =(Ap - BpKe)e

Remark 1. Stability analysis: checking the controllability of the (Ap, Bp), there exists an appropriate Ke to let the poles of Ap — BpKe have negative real part, which mean e tends to be zero as t ? oo.

3.3. Switch mechanism of THS

In subsection 3.2, the model following method is utilized to design the general reconfigurable control law by taking THS into account for the aircraft with some elevators failure. However, it still needs to decide whether to use THS or not, because sometimes it is enough to just use the available elevators to accomplish some flight mission even without the assistance of THS. For the purpose of both better system performance and more effective utilization of the control surface, we propose an improvement coefficient g to decide when the THS should be controlled to generate the extra pitch moment. Firstly, some definitions are presented:

R: Control mission, presented as the command (step input) of pitch angle.

ue: Using only the available elevators in fault-tolerant control.

ues: Using both available elevators and THS in fault-tolerant control.

Ce: Fault-tolerant controller corresponding to ue. Ces: Fault-tolerant controller corresponding to ues. ts90% (R, C, u): Rising-time needed for the pitch angle to arrive at 90% of the step command, given the control mission R, available control surface u and fault-tolerant controller C.

With all the above definitions, a performance improvement coefficient is defined as

's,90%

(R, Ce,

- 4,90% (R, C

's,90%(R, Ces, Mes)

'Ces' Ues)x 100%

The essential meaning of the improvement coefficient g is how much will be improved due to involving THS in the reconfigurable control. When some elevators fail, by predicting g and comparing it with g0 given by the system, the FTC system can decide to use THS not or and then choose the appropriate controller. The switching process is shown in Fig. 3.

Fig. 3 Diagram of switching mechanism.

This switching process can also be described as follows: If g P g0, the controller Ces will be selected; Else, the controller Ce will be selected.

Remark 2. In reality, the determination of any parameter in flight controller should be on the basis of many simulations and experiments for a specific aircraft. Here, the stored threshold value g0 can be pre-designed according to the flight condition, required performance and other composite factors.

4. Simulation results and discussion

4.1. Simulation setup and controller parameters

The reconfigurable control strategy is implemented on a high fidelity simulation model on the B747-200, and the airplane model is trimmed at straight and level flight with a flight condition of VT = 0.8Ma and at 40000 ft (1 ft = 0.3048 m) height. The detailed data of linear model derives from30:

-32.02 -2.8

0.00105 -0.42878 0.0003248

-0.00276 0.0389 0

-0.0650 -0.317 771 0.000193

0 0 1 B =[ 1.44 -17.9 -1.16 0 ]T 0.72 -8.95 -0.58 0 #T 3.48 -43.3 -2.46 0 0.36 -4.47 -0.29 0 3.48 43.4 2.46 0

where the three control matrices corresponding to the three situations: B is with no faulty elevator, B2 is with two faulty elevators and B3 is with three faulty elevators. The corresponding control laws are given as

[-0.0002 0.0009 -12.7755 -42.8005 ] Kc — 31.6228

K2e = K2m = K3e = K3m =

-3.3 x 10-4 1.6 x 10-3 -8 x 10-5 4.1 x 10-4 -0.0004 0.0018 -25.6 0 0 0 -3.0 x 10-4 1.6 x 10-3 0 0 -0.0007 0.0034 -51.1 0 0 0

-19.1 -20.9 -7.3 -6.3 -85.6 63.2" 0 0 -5.6 -5.9 0.05 -3 -3 0.05 -171.2 126 0 0

0.1 0.15

where K2e and K2m are designed for system with two faulty elevators by using THS; K3e and K3m are designed for system with three faulty elevators by using THS. When FTCS is switched to controller without utilizing THS, the second row of K2e and K3e are set to be zero.

4.2. Longitudinal AFTC simulation with pitch angle command of 10°

In this simulation, the pitch angle command from flight control system is set as a plus signal with amplitude of 10°. Firstly, the simulation results with two faulty elevators are carried out and shown in Fig. 4.

Fig. 4(a) shows the compared step response results with different control situations. Then Fig. 4(b) and 4(c) give the

dynamic deflection of elevators and THS, respectively. From Fig. 4(a), we can find that with two faulty elevators, the proposed AFTC strategy (whether using THS or not) can achieve the same performance to track the desired pitch angle as under normal condition and the stability is guaranteed. Furthermore, in such condition of two faulty elevators and small pitch angle command, the effectiveness of THS is not obvious.

Then, with the same pitch angle command, the simulation runs under another failure model with three faulty elevators, and the simulation results are shown in Fig. 5. In this simulation, the failure model is more serious than the last one, thus, from Fig. 5(a), we can find that the dynamic tracking performance is degraded, but the system stability is still guaranteed. Fig. 4(b) and 4(c) give the dynamic deflection of elevators and THS, and the effectiveness of THS is also not obvious.

To sum up, the proposed AFTC algorithm can achieve desired longitudinal control performance and guarantee system stability with two or three faulty elevators. Moreover, it is validated that THS is less helpful to longitudinal control when the pitch angle command is small.

4.3. Longitudinal AFTC simulation with pitch angle command of 20°

In this subsection, in order to further verify the fault-tolerant control performance of the proposed algorithm, the simulation runs with bigger pitch angle command which is a plus signal

15 ^ 10

> <L> 0

0.6 0.2

H -0.2

---Two failed elevators, with THS -Two failed elevators, without THS

j--Command /-Normal V

5 10 15 20 25 30 Time (s) (a) Step response of pitch angle

-Normal ---Two failed elevators, with THS -Two failed elevators, without THS-

0 10 15 20 25 30 Time (s) (b) Elevator deflection

5 10 15 20 25 30 Time (s) (c) THS deflection

Fig. 4 Simulation results with pitch angle command of 10° when two elevators fail.

---Three failed elevators, with THS -Three failed elevators, without THS

/»-Normal V.

10 15 20 25 Time (s) (a) Step response of pitch angle

-Normal ---Two failed elevators, with THS -Two failed elevators, without THS ■

10 15 20 25 30 Time (s) (b) Elevator deflection

10 15 20

Time (s) (c) THS deflection

Fig. 5 Simulation results with pitch angle command of 10° when three elevators fail.

with amplitude of 20°, and the simulation results are given in Figs. 6 and 7.

Figs. 6(a) and 7(a) give the compared dynamic responses with two and three faulty elevators, respectively. Compared with Figs. 6(a) and 7(a), we can find that with the faulty elevators, the reconfigurable control law is effective since the response of the airplane can finally track the desired output (step) without steady error. It seems no remarkable difference with or without THS when there are two faulty elevators; however, when the number of faulty elevators increases to three, which means the failure is more serious, AFTC with THS acts quicker than that without THS, but both of them will act slower than the normal system. The reason for the difference of step response between FTCS with two and three faulty elevators can be explained by Fig. 7(c), which shows that when the aircraft elevator failure is more serious, The response of THS will be more greater to compensate much more pitch moment needed for aircraft pitch control with performance degradation. In some way, THS is used to reduce the degree of performance degradation by providing some extra pitch moment slowly with a deflection rate at 0.2 (°)/s, as shown in Fig. 7(c).

Totally speaking, the proposed AFTC algorithm can effectively compensate the fault of elevators and guarantee the stability of the longitudinal control system after elevator failures occur.

OJ) §

a 10 E

j> 0 w

---Three failed elevators, with THS -Three failed elevators, without THS

ir 1 /-Command / J -Normal V

i 10 15 20 25 Time (s)

(a) Step response of pitch angle

-Normal ---Two failed elevators, with THS -Two failed elevators, without THS

10 15 20 Time (s) (b) Elevator deflection

10 15 20

Time (s) (c) THS deflection

Fig. 7 Simulation results with pitch angle command of 20° when three elevators fail.

4.4. Simulation and analysis of effectiveness of THS

The extra pitch moment from THS can contribute to improve system performance, depending on not only how serious the elevator failures are, but also how much pitch moment the system needs. As shown in Fig. 8, when the flight mission changes, the improvement performance contributed by THS will be clearly different.

We define the magnitude of the tracking pitch angle as the flight mission, from 1° to 30°. The rising-time ts 90% (R, C, u) is concerned here to evaluate the quickness of FTCS. Fig. 8 shows that whether THS is used or not after two elevators'

Fig. 6 Simulation results with pitch angle command of 20° when two elevators fail.

Fig. 8 Relationship between g and step command with elevator failure.

failure, there is no obvious improvement on response quickness as the magnitude of command increases (dashed line in Fig. 8). This is because the two remaining elevators under good condition can provide most of the pitch moment, leading to little effect of extra pitch moment from THS. However, when three elevators fail (solid line in Fig. 8), with the assistance of THS, the rising-time will be shortened to much greater extent than no THS assistance as the pitch command increases. For example, when the pitch command is set to be 30°, the rising time will be reduced from 10.2 s to 7 s if the THS is taken into account in the proposed AFTC, which is a remarkable improvement on response quickness. According to Fig. 8, the improvement coefficient g can be seen as an indicator of the criterion for the switching condition.

5. Conclusions

An AFTC strategy based on model following technique has been proposed in this study for civil aircraft longitudinal control after some elevators fail. By introducing the performance improvement coefficient, AFTC can provide the appropriate controller under elevator faults to achieve the good performance. Implementing the control strategy on B747, the control law is able to track step command preciously without the steady-error. When less two elevators fail, there is no obvious difference whether to utilize the THS in FTCS or not with the increment of amplitude of step command. However, when three elevators fail, THS can play an important role in improving the response quickness as the amplitude of command (step input) increases. It is reasonable to use the performance improvement coefficient as a criterion and to design a switching mechanism to decide when to take advantage of THS, so that a good balance between better system performance and less control complexity will be achieved.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 51305011), the National Basic Research Program of China (No. 2014CB046402) and the 111 Project of China.

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Wang Xingjian received the Ph.D. and B.E. degrees in mechatronics engineering from Beihang University, China, in 2012 and 2006. From 2009 to 2010, he was a visiting scholar in the School of Mechanical Engineering, Purdue University, West Lafayette, IN, U.S. He is currently with the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His research interests are nonlinear control, active fault tolerant control, fault diagnostic and fault prognostic.

Wang Shaoping received the Ph.D., M.E. and B.E. degrees in mecha-tronics engineering from Beihang University, China, in 1994, 1991 and 1988. She has been with the Automation Science and Electrical Engineering at Beihang University since 1994 and promoted to the rank of professor in 2000. Her research interests are engineering reliability, fault diagnostic, prognostic and health management, as well as active fault tolerant control.

Yang Zhongwei received his M.E. and B.E. degrees in mechatronics engineering from Beihang University in 2015 and 2012, respectively. His main research interests are active fault tolerant control, and fault diagnostic.

Zhang Chao received his B.E. and Ph.D. degrees in mechatronics engineering from Beihang University in 2008 and 2014, respectively, and then became an instructor there. His main research interests are reliability testing, performance degradation, and fault diagnosis of hydraulic systems.