CHINESE

JOURNAL OF

AERONAUTICS

Chinese Society of Aeronautics and ronautics 'Beihang 'University

Accepted Manuscript

Full length article

Active fault tolerant control for vertical tail damaged aircraft with dissimilar redundant actuation system

Wang Jun, Wang Shaoping, Wang Xingjian, Shi Cun, Mileta M. Tomovic

PII: DOI:

Reference:

S1000-9361(16)30103-0 http://dx.doi.org/10.1016/j.cja.2016.08.009 CJA 669

To appear in:

Chinese Journal of Aeronautics

Received Date: Revised Date: Accepted Date:

13 October 2015 28 March 2016 13 May 2016

Please cite this article as: W. Jun, W. Shaoping, W. Xingjian, S. Cun, M.M. Tomovic, Active fault tolerant control for vertical tail damaged aircraft with dissimilar redundant actuation system, Chinese Journal of Aeronautics (2016), doi: http://dx.doi.org/10.1016/j.cja.2016.08.009

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Chinese Journal of Aeronautics 28 (2015) xx-xx

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Chinese Journal of Aeronautics

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Final Accepted Version

Active fault tolerant contro. for vertical tail damaged aircraft with dissimilar redundant actuation system

Wang Juna, Wang Shaopinga, Wang Xingjiana*, Shi Cuna, Mileta M. Tomovicb

aSchool of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China b College of Engineering and Technology, Old Dominion University, Norfolk VA 23529, USA Received 13 October 2015; revised 28 March 2016; accepted 13 May 2016

Abstract

This paper proposes an active fault-tolerant control strategy for an aircraft with dissimilar redundant actuation system (DRAS) that has suffered from vertical tail damage. A damage degree coefficient based on the effective vertical tail area is introduced to parameterize the damaged flight dynamic model. The nonlinear relationship between the damage degree coefficient and the corresponding stability derivatives is considered. Furthermore, the performance degradation of new input channel with electro-hydrostatic actuator (EHA) is also taken into account in the damaged flight dynamic model. Based on the accurate damaged flight dynamic model, a composite method of linear quadratic regulator (LQR) integrating model reference adaptive control (MRAC) is proposed to reconfigure the fault-tolerant control law. The numerical simulation results validate the effectiveness of the proposed fault-tolerant control strategy with accurate flight dynamic model. The results also indicate that aircraft with DRAS has better fault-tolerant control ability than the traditional ones when the vertical tail suffers from serious damage.

Keywords: Fault-tolerant control; Model reference adaptive control; Linear quadratic regulator; Vertical tail loss; Nonlinear aircraft model; Electro-hydrostatic actuator; Dissimilar redundant actuation system

Corresponding author. Tel.: +86 10 82338917 E-mail address: wangxj @buaa.edu.cn

nail address.

Introdu

1. Introduction

Structural damage to an aircraft, like the damage/loss of a vertical tail, can lead to loss of controllability, which would create a challenging situation for the pilots.1-3 An example of such a situation, is the disaster that involved the Boeing 747 freighter aircraft that crashed in Mount Osutaka in 1985, with no one survived (520 fatalities). In this particular case, the aircraft lost the vertical tail and the hydraulic pipelines were pulled apart. This damage caused significant loss of controllability and, next to that, structural changes, which led to the crash. Another such similar example was 2001-A300, vertical tail loss, 265 fatalities. Such failures were likely to be survivable, if given correct control input and a wise trajectory. However, there were no effective measures for these aircraft with traditional centralized hydraulic actuation system (HAS) in such extreme situations, since the vertical tail loss would pull apart the hydraulic pipelines and lead the aircraft to lose pressures to actuate. Fortunately, the modern civil aircraft are developing towards the trend of being powered electrically more and more. Electro-hydrostatic actuators (EHA) have been applied in aircraft together with traditional centralized HAS, which produces dissimilar redundant actuation system (DRAS). 45 Consequently, there has been a growing interest in new-type aircraft with this kind of new actuation system. The commercial aircraft, A380, A350 and A400M of Airbus Company have adopted

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DRAS of 2H/2E type. Researches about DRAS are being done as one effective measure to further enhance the flight safety.6 These researches indicate that aircraft with DRAS have potential fault-tolerant capability to respond to some extreme situations. Therefore, it is desirable to develop fault-tolerant mechanisms for aircraft with DRAS that can assist the crew in some severe situations.

Outer mold line changes due to the damage can result in nonlinear and/or non-symmetric mass properties, aerodynamics, or control characteristics.7 Though it is difficult to analytically estimate or predict such characteristics or their impact, it is still very necessary to study the relationship between the damage degree and the dynamic model to service the fault-tolerant control (FTC) strategy design. The general effects of vertical tail damage on directional characteristics are similar in nature to those seen in the pitch axis from stabilizer damage, namely, a reduction in static and dynamic stability. Much earlier work has been done to study the damaged aircraft modeling and the fault-tolerant strategy.8-10 In these researches, the researchers studied the damaged aircraft model with vertical tail loss with not very serious damage using Boeing-747 100/200 data and regarded the input channel can continue to work. However, it is highly possible for the aircraft only with centralized HAS to lose the input channel when suffering from serious vertical tail loss. Besides, in these researches, the damage-induced aerodynamics characteristics change is expressed as a linear scale of the maximum damage degree, the accurate nonlinear relationship between the damage degree and its corresponding stability and control derivatives has not been studied.

For the research about vertical tail loss,10 since the researchers studied the not very serious damage degree, three passive fault-tolerant control (PFTC) methods: quadratic stabilization, guaranteed cost control and quadratic cost control with robust pole placement were compared. The conclusion is that guaranteed cost control with robust pole placement can have a better control performance when the damage is less than 10%. However, when the damage degree exceeds a critical number, those PFTC methods would be no longer applicable. An active fault-tolerant control (AFTC)11-15 method should be chosen to respond to extreme situations. Many researchers developed effective control methods to cope with FTC problems of complex system. In Li and Yang's latest research work,16 an adaptive fuzzy decentralized control method was used to solve the FTC problem of large-scale nonlinear systems with actuator faults and unknown dead zone; in their another work,17 a robust fuzzy adaptive control method was used to solve the FTC problem of large-scale nonlinear systems with mismatched uncertainties and actuator faults. Both researches indicate the effectiveness of adaptive technique as one important factor of the control method. To compensate for the serious failures of aircraft, Stengel and Huang studied reconfigurable control using proportional-integral implicit model following method very early.18 Bodson and Groszkiewicz developed a multivariable adaptive algorithm for reconfigurable flight control system.19 Boskovic and Mehra developed an adaptive control method for a tailless advanced fighter aircraft under wing damage.20 The later researches, e.g. in Lavretsky's research, an composite model reference adaptive control (MRAC) method was developed by integrating the classical model following method and adaptive control.21 These researches indicate that MRAC can be an effective FTC method and it is necessary to highlight that in flight control system the determination of control law parameters should be more efficient.

This paper studies the civil aircraft with DRAS and focuses on the modeling of damaged aircraft in vertical tail loss situation and developing FTC strategy. A damage degree coefficient based on the effective vertical tail area is introduced, and then the nonlinear relationship between the damage degree and its corresponding stability and control derivatives are studied. In this way, a flight dynamic model for a damaged aircraft is developed to account for various damage degrees that result in changes to aerodynamics. Furthermore, even the hydraulic systems lose pressure when the pipelines are pulled apart due to vertical tail loss, EHA can continue to actuate to stabilize the dynamic model. Therefore, the performance of EHA is also modeled in the dynamic model. Based on the accurate modeling, an AFTC strategy, using MRAC composed of linear quadratic regulator (LQR) technique,22'23 is developed. In this way, the control law parameters can be determined by LQR method while the fault-tolerance can be guaranteed by MRAC method. Simulation results illustrate the necessity of accurate modeling and effectiveness of designed FTC strategy.

2. Modeling of damaged aircraft

It is a standard practice to linearize the model around a certain steady flight operating point. When the aircraft suffers from vertical tail damage, only some lateral-directional parameters change, therefore, only the lateral-directional model of the aircraft was considered in this paper. Using the early research work,1024 the aircraft model under normal conditions can be represented as follows:

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where the state variable vector x = [3, p, r, 0]T, in which 3 is the sideslip angle, p the roll rate, r the yaw rate, and 0 the roll angle; the system corresponding matrices are

Yß Yp.

Lß IX + CNß L f + IxNP X

Nß I' + y + I'xLp

h, IX + CN, L, -y+ IX

N I' -+Ch, N, , — + IL, ZX Sr

0 0 0 1"

1 0 0 0

-+ IZxLr

tan#„

g0cos&0

For the system matrices, the nomenclature for all the parameters in matrices A and B is the same as in Li and Liu's research work.10 Where I' = (IxIi - I2a )/li , I' = (IxIi - I2a )llx , I'a = Ia/(IJZ - I2a) ; Ix and I

are inertia moments about (x, z) axes respectively; Ix is the inertia product due to (x, z) axes; Yp, Yp and Yr are the side aerodynamic forces in three stability axes respectively; Lp , Lp and Lr are the rolling moments in three stability axes respectively; Nfi , Np and Nr are the yawing moments in three stability axes respectively; Ys and Ys are side aerodynamic forces provided by ailerons and rudders respectively; Ls and Ls are the rolling moments provided by ailerons and rudders respectively; Ns and Ns are the yawing moments provided by ailerons and rudders respectively; m is the total mass of the airplane; u0 is the reference flight speed; d0 is the reference angle of climb; g0 is the acceleration due to gravity.

uH =[£a, Sr ]T is the control input vector, and Sa the control input provided by ailerons, Sr the control input provided by rudders. The output matrix C indicates that sideslip angle 3 and roll angle 0 are the system output.

In this section, the aircraft dynamic model has been well developed. An accurate model can be obtained by considering the nonlinear damage of vertical tail and the performance degradation of EHA.

2.1. Modeling of nonlinear damage

2.1.1. Modeling of vertical tail damage

For aircraft suffering from vertical tail damage, as shown in Fig. 1, the fracture shape caused by vertical tail loss is irregular and some relative assumptions are made for the proceeding of the research.

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Fig.1 An aircraft with irregular partial vertical tail loss.

Assumption 1. The vertical tail shape is regular trapezium and the loss part of vertical tail is equivalent to regular trapezium.

Based on Assumption 1, a parameter ju is introduced to define the damage degree. Definition 1. Vertical tail damage coefficient ju : the ratio of lost effective area in all eff s of the vertical

With the above definition, jue [0,1] can be used to represent different dan cases, especially, j = 0 represents that the flight model is in normal condition without damage. In order to n the nonlinear relationship between ju and all the corresponding stability derivative deviation, the fundam equation of vertical tail derivative in Eq. (2) is used as

V S ref

tan2 Anax ß2

where is the damaged vertical tail derivative; ¡3l = 1 - Ma , F = 1.07 (1 + d/b)

number, b the height of vertical tail and d the fuselage diameter; Sout and Sref

with Ma the Mach are exposed area and

reference area respectively; A = b2/Slf is the aspect ratio of vertical tail; Amax t is the vertical tail sweep of string position of airfoil thickness; and n is the efficiency of airfoil.

Based on Assumption 1 and Definition 1, several geometric parameters in the case of vertical tail partial loss can be obtained. As shown in Fig. 2, ct and cr are the vertical tail tip chord length and the vertical tail root chord length respectively. The exposed area of vertical tail can be obtained as that in Eq. (3).

(c, + Cr) b

Fig. 2 Regular damage and relative parameters of vertical tail loss.

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When the vertical tail damage degree is u , the vertical tail tip chord length changes from ct to ct (u); meanwhile, the height of vertical tail deduces by b(u), also the aspect ratio of vertical tail changes into A(u); considering all these factors, the nonlinear relationship between u and Cy3 can be obtained based on trapezoid area equation. As shown in Fig. 2, the lost area of vertical tail can be represented as

= 2 (ct + Ct (/)) b (/) (4)

and the rest area of the vertical tail should be

(1 -/) S0ut = 2 (Cr + Ct (U))(b - b (/)) (5)

(ct + Cr )u

Combining Eqs. (3) and (4), b (u) can be obtained as b (u) =-—7b, thei

Ct + Ct (m)

tail height can be represented as

bleft = b - b (/)= b

1 (Ct + Cr )M

ct + ct (m)

:n the effe

ective rest vertical

Combining Eqs. (5) and (6), the representation of the vertical tail tip chord length can be obtained as

2 (1 -m) Ct + 2MCr

Ct (M) =\l-"-Sout - CtCr

Then the aspect ratio of vertical tail when the

A (m) =

Sref Sout + Sout (1 M)

l suffers from u degree damage can be obtained

Finally the nonlinear relationship between u and Cy can be obtained using the above equations as

(m) = -

Cy„ (M) -

2nA (m) ' Sout (1 -M) ' V1+bf i

_ ( Sref - Sout ) + Sout (1 -M) _

2+i 4 + A2 (M)ß2 4 + I1 f, tan2 4maxt ^ 1 + ß2 V ^ J

In order to simulate the nonlinear relationship between u and Cy^, this paper uses vertical tail characteristics of Boeing-747 and its corresponding parameters are chosen as: Sout = 77 m2 , Sref = 87 m2 , ¡3 = n/16, d = 6.5m, b = 9.8m, A^ t = 40°, n = 0.95, ct = 4m, cr = 11.7m.

The relationship between u and Cy3 is shown in Fig. 3, in which Cy3 decreases gradually to 0 as u changes from 0 to 1. In this process, the relationship between u and Cy3 presents strong nonlinearity. The

absolute difference AC,, vs u is shown in Fig. 4, in which AC,, reaches the maximum value when

I y3 I I y3 I

the damage degree u = 0.9. Based on this nonlinear function about u and Cy3, one more precise vertical tail damaged aircraft model can be obtained.

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Vertical tail damage Fig. 3 Function relationship between C and JU .

Fig. 4 Absolute value difference of vs JU

2.1.2. Corresponding dimensional aerodynamic derivatives

In the case of aircraft suffering from vertical tail damage, there are also other damage-induced stability and control derivatives. Using Hitachi' s research results,8 the relationships between C (j) and the rest corresponding derivatives can be obtained as

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Clß(M) = Cyr C") =

Cn, (A =

c, (m) =

Cyr(M) =

cnr(M) = c,(m) =

—cosa+ —sina

v bw bw

—cosa+—sina

Cyß (A)

—cosa—-sina

v bw bw

C,. A)

—Y (lv cos a + zv sin a)( zv cos a- lv sin a)

--(lv cosa+ zv sina)

-2- (lv cosa+ zv sina)2 b„,

—^ (lv cosa+ zv sina)( zv cosa-lv sina)

The nomenclature for all the parameters in Eq. (10) and the following Eq. (11) is the same as in Hitachi' s research results.8 Where Cn^(u) is the weathercock stability derivative; C ,^(u) is the dihedral effect;

Cy (u) is the side force-in-roll derivative, Cn (u) is the cross derivative of rolling-caused yawing moment; C, (u) is the damping-in-roll derivative; Cy^(u) is the attack angle-caused derivative; C^(u) is the damping-in-yaw derivative; C^ (u) is the cross derivative of yawing-caused rolling moment; bw is the wing span; ,v and zv are horizontal and vertical location of the aerodynamic center of the vertical tail; a is the attack angle of the zero lift line.

The affected stability and control derivatives indicate that the vertical tail damage would have significant impact on lateral and directional dynamic behavior. The corresponding changing aerodynamic derivatives in the lateral and directional dynamic model are listed as

Y,(U) = 2 Psw Cyf(u)

Yr (u) = 4PbwSwCyr (u)

Yp (u) = 4 Pbw Sw Cyp (u)

Lp(u) = 2 Pbw Sw Clf(U)

Lr (u) = 4PKswClr (u) (11)

Lp (u) = 4PblSwClp (u)

Np(u) = 1 Pbw Sw Cn^(u) Nr (u) =1 Pbw Sw c (u)

Np (<u) = 4Pb2wSwCn, (A)

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where p is the air density; Sw is the wing reference area. 2.2. Design of performance degradation input function

In this research, the civil aircraft with DRAS was studied. The structure of DRAS is shown in Fig. 5, where uh

and ue are the input signals for HA and EHA respectively, iv the electrical signal for the servo valve, and 0d

the deflection angle of the control surface. When the vertical tail suffers from serious damage, the vertical tail loss will pull apart the hydraulic pipelines, and the whole hydraulic system will lose pressure. In this case, all control surfaces actuated by HA system would be in loose floating state, and then a catastrophic consequence may occur. Fortunately, if the new-type civil aircraft with DRAS can launch EHA to actuate some main control surfaces quickly, the aircraft can survive using this new type actuation system.

Fig. 5 Dissimilar redundant actuation system composed of HA and EHA.

HA and EHA are different in response performance. EHA responds more slowly than HA. Therefore, the response performance degradation of EHA should be considered. In this section, the performance degradation input function was designed according to the transfer functions of HA and EHA.

2.2.1. Transfer funct

HA and EHA

In the early research, the force fighting problem of DRAS has been studied, similarly with the research work, the variable vector of DRAS was defined as

DRAS Z[ X11' x12

Ph> xv' xe' Ve, Pe' ^e ]

'23, x24

]T (12)

where xh and xe are cylinder piston displacement of HA and EHA respectively; vh and ve are cylinder piston velocity of HA and EHA respectively; Ph and Pe are loading pressure of HA and EHA respectively; xv is the servo valve spool displacement of HA; and oe is the motor speed of EHA.

Based on the research work,25 the state space model of HA can be obtained as

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Xn X12

Bh + B + Bd x — h e d x +_ x12 x12 ^

4Eh Ah

—__h h v .

13 X12

4 Eh Kci x + 4EK x

x13 + T, x14

where Bh, Be and Bd are equivalent damping parameter of hydraulic cylinder piston of HA and EHA, and the control surface respectively; mh, me and md are equivalent mass of hydraulic cylinder piston of HA and EHA, and the control surface respectively; Ah is the area of HA hydraulic cylinder piston; Eh is the volumetric modulus of elasticity of HA; Vh is the total volume of the HA hydraulic cylinder; Kcl is the sum of flow pressure and leakage coefficient of the HA hydraulic cylinder; Kq is the flow change coefficient; Kv is the proportionality coefficient of electro-hydraulic servo valve; Tv is the servo valve time constant.

Using Laplace transformation, the transfer function of HA based on its state space model can be obtained as X h (i)

G (s) —-

ah s4 + bh s3 + ch s2 + dh s

— ( mh + me + md )VhTv

^ 4 Eh Ah

bh — [KT (Bh + Be + Bd) + (4ettKci + Vh)

•( mh + me + md)]/ 4Eh Ah Ch — [(4EhT Kci + Vh) (Bh + Be + Bd) + 4Eh Kci •( mh + me + md)]/ 4E Ah +T Ah (Bh + Be + Bd) kci + A[

dh —

r all the whic

The nomenclature for all the parameters in the equations above is the same as in the research work. Then the dominant pole sH = aH which determines the response performance of HA can be obtained. Still based on the research work,25 the state space model of EHA can be obtained as

. — - Bh + Be + Bd _

X22 X22 + X23

mh + me + md mh + me + md

4EeCei 4EeVp

Y —__e e V__e V I__e p r

X23 TX22 xX2^ xX24

.Bm J m

J m Re

14 X14 +

X21 X22

X24 —

where Ae is the area of EHA hydraulic cylinder piston; Ee is the volumetric modulus of elasticity of EHA; Ve is the total volume of EHA hydraulic cylinder; Cel is the total leakage coefficient of EHA hydraulic cylinder; VP is the pump output of EHA; Jm is the total moment of inertia of motor and pump; Bme is the simplified equivalent damping coefficient of the motor; Km is the electromagnetic torque constant of the motor; Re is the is the armature resistance of the motor.

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Using Laplace transformation, the transfer function of EHA based on its state space model can be obtained as X e ( s )

G ( 5 )-

K m/ R

u2 ( 5 ) ae s4 + be 53 + ce 5 2 + de 5

J mVe ( M + me + Md )

4EeVPAe

be = JmVe ( Bh + Be + Bd )/4EeVp A + [( 4Ee Jm Cei

+ BmeVe ) (M + M + md )]/4EeVp Ae

^e = ( 4 Ee J m Cd + BmeVe )( B„ + Be + B, )/4 E&Vp A + m +

All ey ^ e d )l ^P Ae

+ (VP2 + BmeCel )(M + M + M, )/VpA + J m Ae/Vp

( Bme Cei + Vp2 )( B„ + Be + B, ) Bme A(

The nomenclature for all the parameters in the equations above is also the same as in the research work.25 Then the dominant pole 5E = aE which determines the response performance of EHA can be obtained.

2.2.2. performance degradation input function

I EHA, a s

In order to compare the response performance of HA and EHA, a signal with the same step for HA and EHA was chosen to be used in simulation. As shown in Fig. 6, EHA responds more slowly than HA.

Fig. 6 Response performance comparison of HA and EHA.

om the transfer functions of HA and EHA, two dominant poles sH = aH and sE = aE which determine the esponse time of HA and EHA respectively can be obtained. Then the degradation performance parameter can be chosen as Aa = |aH - aE|, and then the performance degradation input function can be obtained as

—Aat

Ur; - e

In this performance degradation input function, uH is the input of HA before the aircraft suffers from vertical tail loss, since the dynamics of actuator on control surface is much faster than the system dynamics, which means the control surface deflection bounded in the limit of amplitude can be considered equal to the command generalized by the controller without delay. However, when the aircraft suffers from vertical tail loss, only EHA provides effective measure for the aircraft to actuate the main control surfaces and the response performance degradation e-Aat should be considered.

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2.3. Accurate model of vertical tail damaged aircraft

When the aircraft suffers from vertical tail damage, the corresponding parameters and the input channel all change. Based on the analytical work in the above two sections, the final accurate damaged model can be obtained as

[x = A( jh) x+B ( jh) ue + d 1 y=Cx

where the state variable vector is still x = [—, p, r, ces are changed into

; d is the external disturbance, and the system matri

A ( M ) =

Yß(M)

-p (m) ix

np (m)

+ I'xNP (M)

+cl (m)

L (m) IX

Nr (M)

B ( m ) =

-y+KN,

Cyß(M)

k (o)|

+1 : L C(M) ( N,

Cyß (0)

g0cosd0

+ I'xNr (M) 0

+ CL (M)

id the syst

where the elements of matrix A (p) are calculated using the nonlinear damaged aircraft modeling method which is proposed in Section 2.1. The elements of the second column of matrix B (p) change with the damage degree of the rudder, since the shape of the rudder is also regular trapezium which is proportional to the vertical tail, the

Cy (m)

damage rule is the same as the vertical tail. Therefore, a new damage coefficient ,—--,, which is also based on

K (°)|

the fundamental equation Cy- (m) , was constructed to obtain the damage matrix B (p). The input channel was replaced by the performance degradation input function as shown in Eq. (19).

^-Aa? -Aat \ £ c 1T

= e uh = e [Oа• yr ]

3. Fault-tolerant control strategy

The structure of the FTC strategy is shown in Fig. 7. Under normal condition, the fault detection and isolation (FDI) mechanism would obtain healthy result of the aircraft, and then the baseline control parameters would be solved by the LQR regulator. In this situation, HA systems actuate to maintain desirable flight attitude. Since this paper focuses on the development of AFTC strategy and there are many mature techniques to obtain the fault information, when the vertical tail of the aircraft suffers from partial loss, the damage degree can be assumed to be determined precisely by the FDI mechanism. The corresponding reconfigurable control law would be solved by the LQR regulator. Based on the information from FDI mechanism, the switch mechanism would change the baseline

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control law into reconfigurable one and launch EHA system for the fault-tolerance of the damaged aircraft. The design steps of the FTC strategy are shown in this section.

Fig. 7 Structure of FTC strategy

3.1. Baseline controller design

When all the control surfaces are in good condition, for the plant, a baseline controller was designed according to the following principles: The baseline controller should guarantee the stability of the whole closed loop system; the roll angle can track the given command r, (t), the tracking error e(t) = r, (t) -,(t) ^ 0 ; meanwhile, the sideslip

angle f(t) ^ 0°.

In order to satisfy the design principles, a closed loop feedback control law was designed in good condition as Uh = Kxx + K, J ( , - ,) dt + Kfl J ( 0 - f ) dt (20) The specific controller structure is shown in Fig.

Fig. 8 Lateral-directional baseline control law.

If the control command was chosen as r = [ rp, r-0 ] and the plant output was chosen as y = [0, fijT , then their error e =r - y can be obtained. In order to eliminate the error, a control variable xc e R2x1 was introduced, and the control variable system should be Xc=Ac xc+Bc (r - y). Since the plant output is the controlled object variable vector and the plant follows the control command vector, then the matrices of the control variable system can be chosen as Ac =0 and Bc =I. Finally an augmented system composed of the plant and the control variable system can be obtained as

x " A 0" x ' B " 0 " ' d "

= + UH + +

[ x c ] [—C 0] [ xc ] [ 0 ] r [ 0 ]

y = 0 = [C 0]

For this augmented system, the system matrix is A =

—C 0

and the control matrix is fi„„ =

Then the controHabnky matrix Ccon = [BmV AmgA^BmV A^BmV A^Bmi, A^Bmi] can be obtained. Using the matrices data in Li and Liu's research work,10 Ccon can be checked as row full rank. Therefore the controllability of the augmented system can be guaranteed. And then, a control law can be designed for poles' assignment so that the closed loop system poles can guarantee the system stability. After poles assignment by the

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control law form uH =Kx x +Kc xc.

the closed loop poles of system can have negative real part. The control law form in Eq. (20) can be obtained by defining Kc = [K0, Kp ] . With the control law Eq. (20), which is essentially a PI controller, the system can track the command with no steady-error. By solving reasonable feedback gain Kx and Kc, the system can have a better dynamic performance. LQR technique was applied in this paper to calculate the feedback gain Kx and Kc . The control law can be conveniently obtained by selecting the state weighting matrix Q and input weighting matrix R, whose objective is to minimize the total control cost, which can be formally stated as

J = 2 O Xaug ßXaug + uH *UH ) dt

where xaug = [x, xc ] and uH is the input vector with HA; the optimal solution to m cost can be obtained by solving the following Riccati Equation:

minimize tl

the total control

^au8 + Ag P - PBaug R-1 Big P + Q = 0

where the matrix P is positive and definite. Then the feedback gain Kx and Kc can be solved as

[ Kx, Kc ] = -R-1 BTug P

3.2. Reconfigurable controller design and adaptive parameter adjustment

•amete

When the aircraft suffers from vertical tail damage, the input channel of HA system would lose its ability to actuate; meanwhile, several stability derivatives change. In this situation, the baseline control law may not be robust enough to tolerate the fault to maintain a graceful flight performance, and may even threaten the flight safety. Therefore, another reconfigurable controller should be designed to stabilize the damaged aircraft model and realize required function reconfiguration. Since the severe loss of vertical tail occurs, the rudder would not continue to provide yaw function and the aileron's roll function can be used to provide the centripetal force to realize the yaw function finally. First, the reconfigurable control gain is resolved using LQR technique and then MRAC method is used to adjust the solved control gain, which can guarantee that the reconfigurable control law is more effective. MRAC mechanism designed in this paper is shown in Fig. 9.

Fig. 9 Mechanism of model reference adaptive control method The reference model can be expressed in the following form:

f x m = Am xm + Bm Um 1 y m = Cm xm

This is a closed loop system controlled by the baseline controller. As an reference model, the system matrices can

be deduced as A

A+BK BK

C„ =

0 0 0 1 0 0 1 0 0 0 0 0

xm = [x> xc L is the

state variable vector, um = rupdate = [r^, rp ]T is the latest command vector and y = [0, ^]T is the system out-

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put. When the aircraft suffers from vertical tail damage, a control variable xc = r - y was introduced as done to the reference model, and then the plant was obtained as

\x p = Ap xp+ Bp "p+ d1 + Br r

k = cp Xp

where Ap

" A (^) 0" B p = " B (^)" , B r = 04x2 , Cp =

_ -C 0_ 6x6 p _ 0 _ r 6x2 _ I _ p 6x2

0 0 0 1 0 0 1 0 0 0 0 0

, d1 =

Since the plant and its reference model have the same dimension, model following method can be applied to designing the fault tolerant control strategy. The error between the state of reference model Eq. (25) and plant Eq. (26) can be defined as eref = xm - xp, and

the control input form was chosen as

Up = K eeref + K m Xm + AUd

where Ke and Km are control gain matrices which need to be solved and AUd is the input compensation which mainly deals with the outside interference d1 in Eq. (26).

Then the differential of state error can be written as

¿ref = X m X p

: Ap ¿ref + (An " Ap ) Xm + Bm Um - Bp Up

- B r - d

Br 'update d1

= ( ^p - Bp Ke ) eref + ( Am - Ap - Bp Km ) Xm

+ ( Bm Um - Br rupdate )-( Bp AU d+ d, )

= ( Ap - Bp Ke ) eref + ( Am - Ap - Bp Km ) Xm

-(BpAUd + d,)

By choosing appropriate gain matrix Km e R2x6 and AUd to satisfy the relationship described by Eq. (27), 0

( Am - Ap - Bp K m

[BpAU d +d1= 0 it leads to

¿ref = ( Ap - Bp Ke ) eref

checking the controllability of (Ap, Bp) and the poles of matrix Ap - Bp Ke have negative real part, the er-

By check: >r ef = x

p' p/ 1 p p e

ror ere

The original feedback control gain matrices Ke and Km were calculated through the plant matrices A (p) and B (p) in theory. AUd was chosen following a certain adaptive adjustment law. Due to the parameter errors

between the plant and the actual system, their closed loop dynamic performance may behave very differently. Adaptive parameter adjustment was used to realize the same dynamic performance, which leads the control law to be more efficient.

The adaptive parameter adjustment law is designed in the following process:

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^ref = X m Xp

= ( A> - Bp Ke ) eref + ( Am " Ap - Bp Km ) Xm

-( Bp AU d ) = ÄAef + Bp ( Ke (t ) - Ke (0)) ^f + Bp ( Km (t ) -Km (0)) Xm - Bp ( AUd (t )- AUd (0))

= ÄAef +Bp^^ref +Bp^Xm + Bp Z

where 0 = Ke (t)- Ke (0), W = Km (t)- Km (0), Z = AUd (t)- AUd (0). By defining Lyapunov function as

2 |>rTef Wref + tr (0T+ WTr2-1W + ZTr3-1Z)] (32)

V = -2

where P„„ e R6

r1 e R6

f2 e R6

and r3 e R6x6 are all positive

the Lyapunov function shown in Eq. (32), the adaptive law can be obtainei "p (t) = K e (t) ^ (t) + K m (t) Xm (t) + AUd

K e (t) = J0' ( eref eTef P^ Bp ^ )' dT + Ke (0)

Km (t) = J0' (XmeTf PadapBpr2 )' dT + Km (0)

AUd (t) = £ (ef Padap Bp r 3 )T dT + AUd (0)

ined as

definite s;

iymmetric matrices. Based on

where Ke (0), Km (0) and AUd (0) are the initial values; Ke (t), Km (t) and AUd (t) are the current moment value obtained through the designed adaptive law.

4. Simulation study

4.1. Flight conditions and data T,

To verify the modeling and the effectiveness of the proposed method in this paper, it was used on the Boeing 747 lateral-directional model. The given model is in steady flight of certain flight condition. The data are listed in Table

Table 1 Flight condition, aircraft parameters and derivatives.

Aircraft property Lateral-directional

Altitude (km) 6.096 Cß (1/rad ) -0.16

Air density (kg/m3) 0.654 (1/rad ) -0.34

Speed (m/s) 205.1304 C lY (1/rad ) 0.13

Wing area (m2) 510.967 C h, (1/rad ) 0.013

Wing span (m) 59.7408 C lh (1/rad ) 0.003

Wing mean chord (m) Weight (kg) 8.3210 288771.723 Cnß CP (1/rad ) (1/rad ) 0.16 -0.026

Ix (kg-m2) 24.6759 x106 Cn (1/rad ) -0.28

Ly (kg-m2) 44.8776 x106 (1/rad ) -0.0018

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!a (kg*m2) 67.3841X106 (1/rad ) -0.100

Iz (kg-m2) 1.3151X106 (1/rad ) -0.90

Air velocity (m/s) 205.1304 0

Thrust (N) 43903.734 0

Air density (kg/m3) 0.654 0

Pressure ratio 0.4695 Cys, 0.12

4.2. Simulation results

4.2.1. Simulation for effectiveness of baseline control law

First, the baseline control law parameters was solved using the proposed fault-tolerant strategy and LQR technique, and then its effectiveness was verified in normal conditions. The specific expressions for system coefficients A and B were given as

-0.1068 0 -3.5276 -0.8442 3.6534 -0.0401 01 0 9.5858 0.2219 0.1030 0.0155 -0.6208 00

-673.0000 0.3088 -0.2479 0.0349

32.1804 0 0 0

The baseline control law parametei

-0.1396 12.5755 31.3745 -4.569

Kbase (0) =

[i dynamo

r matrix

xpression

was solved as

19.5440 -9.3071 -3.6576 -4.4717 3.6576 -9.3071

Fig. 10 shows the system dynamic performance in normal conditions. As shown in Fig. 10(a), a sine wave signal with peak value of +12° was given to the model as the roll command. The roll angel response can track the command well with acceptable time delay, meanwhile, the fluctuation amplitude of the sideslip angel is limited in +0.1° and can be regarded as zero steady state.

be regard

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Fig. 10 System dynamic performance in normal condition.

As shown in Fig. 10(b), when given a step signal of 12° during 10-20 s and 40-50 s as the roll command, during the control process, the roll response curve can track the command well and the sideslip angle can steady at zero value finally.

The above simulation results illustrate the effectiveness of the baseline control law design.

4.2.2. Simulation for comparisons between modeling and fault-tolerant control methods

In this section, three methods were chosen to compare with each other to illustrate the control improvement of the nonlinear modeling method and the effectiveness of the proposed AFTC method. Notations for the three methods were made as follows.

Method 1 (M1): PFTC method (guaranteed cost control with robust pole placement) with linear modeling for the damaged aircraft model in reference.10

Method 2 (M 2): AFTC method (MRAC composed with LQR regulator) with linear modeling for the damaged aircraft model.

Method 3 (M 3): AFTC method (MRAC composed with LQR regulator) with nonlinear accurate modeling for the damaged aircraft proposed in this paper.

The absolute difference | reaches the maximum value when the damage degree ju=0.9, it means the modeling error also reaches the maximum value, therefore, the linear and nonlinear modeling methods were com-

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pared at ^=0.9.

The system coefficient matrices using linear modeling method were obtained as

Äne« (0.9) =

B[inear (0.9) =

9.9433 1.2081 -3.3920 -0.8041 -2.6928 -0.0988

0.2219 0.0155 0

0.9586 0.0103 -0.0621 0

-679.3420 0.1275 -0.0607 0.0349

32.1804 0 0 0

It has been proved that M1 has obvious effectiveness when the damage degree is not very serious. In this paper the damage degree n=0.9 was chosen to verify the control result of the PFTC method compared with the AFTC ones. LMI constraint conditions in Ref.10 were solved to obtain the control parameter matrix of M1 as

'( 0.9) =

-16.6368 3.6721 -12.0535 0.5426 -10.5161 -1.2579 705.1688 -33.4036

The initial control parameter matrix of M2 was solved using LQR regulator as

'( 0.9) =

-5.8781 1.7012 80.8187 -2.0381 -0.3106 -0.0592 51.4982 3.0704 -730.0424 34.2196 0.0592 -0.3106

The system coefficient matrices using the proposed nonlinear modeling method were obtained as

Aonlinear (0.9) =

(0.9) =

5.5883 0.6846 -676.5938 32.1804

-3.4507 -0.8215 0.2061 0

0.0572 -0.0734 -0.0729 0

0 1 0.0349 0

0 0.2219 4.6970 ] 0.0505

-0.3042 0

The initial corresponding fault tolerant control parameter matrix of M3 was solved as T-0 0651 0 144Q 2 6504 0 0422 -0 0

r (0.9) =

-0.0651 0.1449 2.6504 0.0422 -0.0345 -0.3143 1.6655 -0.3998 -65.5610 -2.2763 0.3143 -0.0345

In this section, a sine wave signal and a step signal were used in simulations and the control results of M,, M2 and M3 were compared with each other. Fig. 11 shows the system dynamics performance comparison between PFTC and AFTC methods. The analysis for the simulation results are as follows.

When giving the damaged model a sine wave signal, the output responses using M1 , M2 and M3 are shown in Fig. 11(a). First, the control results of M, and M2 were compared: using M, to control the damaged model, the roll angle can hardly track the command signal and diverge to extremely serious degree, meanwhile, the sideslip angle also presents divergent trend. Air crash would happen in this situation. Simulation results illustrate that it is futile to use M1 to control the damaged model to serious damage degree, and the aircraft would crash due to extreme rolling act. Compared with M,, when using M2, though time delay exists, the roll response can track the command with small sideslip angle mostly. It illustrates that when the damage degree is serious to some certain extent, AFTC method is more efficient and can avoid air crash. Second, the control results of M2 and M3 were

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compared: compared with M2, though the fluctuation amplitude in sideslip response is a little bigger, it is very small and acceptable. Meanwhile, the time delay of tracking performance is obviously shorter since At2 > 2Atj; besides, using M3 can eliminate the wave error Ad . This comparison illustrates that based on the proposed nonlinear accurate modeling model, the control performance for the damaged aircraft model can be improved.

The roll command signal was chosen in the form of steps during 10-20 s and 40-50 s. The output responses using Mj, M2 and M3 are all shown in Fig. 11(b). First, the control results of Mx and M2 were compared: similarly to the sine wave signal condition, using M1 to control the damaged model, the roll angle can hardly track the command signal and diverge to extremely serious degree; meanwhile, the sideslip angel also presents divergent trend. Simulation results illustrate that it is futile to use M1 to control the damaged model when ju=0.9 , the aircraft would crash due to extreme rolling act. Compared with M1, when using M2, the sideslip angle can stabilize near zero value and the roll response does not diverge; however, the tracking performance is still very bad. This comparison illustrates that when the damage degree is serious to some certain extent, AFTC method is very necessary. Second, the control results of M2 and M3 were compared: compared with M2, though there exist fluctuations in sideslip response, the fluctuation amplitude and the frequency are very small and acceptable; meanwhile, the tracking performance of the roll angle is obviously better than that using M2. This comparison illustrates that based on the proposed nonlinear accurate modeling model, the control performance for the damaged aircraft model can be improved.

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Fig. H System dynamics performance using PFTC and AFTC with /U=0.Q .

4.2.3. Simulation for effectiveness of AFTC method in maximum damage condition

In this section, the control effectiveness of the proposed AFTC method in maximum damage condition was verified. The damage degree was chosen as ^^.0, which means the aircraft totally lose the whole vertical tail. The damaged model matrices and the control law matrix were calculated as

A(1) =

B (1) =

11.0600 1.3423 -3.3769 -0.7996 -3.3979 -0.1053

0.2219 0.0155 0

-680.0467 0.1074 -0.0952 0.0349

32.1804 0 0 0

-625.2264 0

-52.5985 0

7.9109 e+ 03 0

-395.5052 0

10.0000 0

)00 0.01

'as verifie . The da

.0134 0

Fig. H shows the system dynamic performance under AFTC control law in maximum damage condition. As shown in Fig. !2(a), when the vertical tail damage degree ,«=L0, using the AFTC method, a sine wave signal was given to the damaged model, the roll angle response can track the command well with acceptable time delay; meanwhile, though the sideslip angle response has sine wave fluctuation, the fluctuation amplitude is limited to be ±0. r and can be regarded as zero steady state.

As shown in Fig. !2(b), when the vertical tail damage degree ,«=L0, using the AFTC method, a step signal of

during !0-20 s and 40-50 s was given to the damaged model as the roll command, the roll angle response can track the command well; meanwhile, though the sideslip angle response has higher fluctuation frequency than before, the fluctuation amplitude is limited to be ±0.2° and finally stabilized at zero value.

The simulation results shown in Fig. H illustrate that even in the most serious damage degree, the proposed AFTC method can maintain the damaged aircraft controllable and safe.

aintain t

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5. Conclusions

aper, the

em dynamic performance under AFTC control law in maximum damage condition.

In this paper, the modeling and control of new-type civil aircraft under vertical tail damage condition have been studied and evaluated.

(1) The fault tolerant control capability of aircrafts with DRAS to respond to extraordinary situations is discussed. A damage degree coefficient based on the effective vertical tail area is introduced, and then the nonlinear relationship between the damage degree and its relevant stability and control derivatives are studied. Meanwhile, the performance of EHA is also modeled in the damaged dynamic model. Considering these two factors, an accurate damaged dynamic model is developed.

(2) The fault tolerant control strategy for the damaged aircraft model with vertical tail loss is studied. Based on the accurate modeling, the fault tolerant strategy, using MRAC composed with (LQR) technique, is developed. In this way, the control law parameters can be determined by LQR method more efficiently and can be adjusted using MRAC to be more precise in fault condition. Simulation results to different damage degrees indicate the effectiveness of the fault tolerant control strategy.

(3) Our further research will focus on the problem of fault degree detection and isolation, and then integrate the FDI technique with the fault tolerant strategy efficiently. Based on this comprehensive technology, our fault tolerant strategies may be more applicable.

■22 ■ Chinese Journal of Aeronautics

Acknowledgements

This work was supported by the National Basic Research Program of China (No. 2014CB046402), the National Natural Science Foundation of China (No. 51575019) and 111 Project.

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Wang Jun is a Ph.D. candidate at school of automation science and electrical engineering, Beihang University, Beijing, China. He received the B.S. degree in mathematics and applied mathematics from Shandong Normal University, Ji'nan, China, in 2011; and then he received the M.S. degree in mathematics from University of Science and Technology Beijing, Beijing, China, in 2014. His main research interests are fault diagnostic and fault tolerant control. E-mail: dwill-wang@buaa.edu.cn

Wang Shaoping received the Ph.D., M.E. and B.E. degrees in mechatronics engineering from Beihang University, Beijing, 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. E-mail: shaopingwang@vip.sina.com

ng from

Beihang Univ Beihang Univ

Engineerin neering rel

Wang Xingjian received the Ph.D. and B.E. degrees in mechatronics engineering from Beihang University, Beijing, 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. E-mail: wangxj @buaa.edu.cn

Shi Cun is a Ph.D. candidate at school of automation science and electrical engineering, Beihang University, Beijing, China. He received the B.E. degree in Mechanical Engineering from China Agricultural University, Beijing, China, in 2014. His main research interests are prognostic and health management and fault tolerant control. E-mail: shicun@buaa.edu.cn

Mileta M. Tomovic received the Ph.D., M.S. and B.S. degrees in mechanical engineering from University of Michigan, Massachusetts Institute of Technology, University of Belgrade, in 1991, 1981 and 1979 respectively. He has been with the mechanical engineering technology department at Purdue University from 1991 to 2008 and served as Chair of Engineering Technology Department in Old Dominion University since 2008. His research interests include design, manufacturing systems and processes, product lifecycle management, system dynamics and control. E-mail: mtomovic @odu.edu

sity si , system dynan