Failure Accommodation for Large Transport Aircraft Using Thrust and Stabilizers

Copyright © IFAC Automatic Control in Aerospace, Seoul, Korea, 1998

FAIL URE ACCOMMODATION FOR LARGE TRANSPORT AIRCRAFT USING THRUST AND STABILIZERS Y. Ochi and K. Kanai

National Defense Academy 1-10-20 Hashirimizu. Yokosuka. 239 Japan

Abstract: This paper describes two topics about failure acconunOdation of large transport aircraft, especially for control surface jam, which can be caused by hydraulic system failure: first, controlling the aircraft with complete loss of hydraulic pressure using engine thrust only; second, taking advantage of slow effectors such as stabilizers and engines to counteract the disturbances caused by the stuck surfaces. Conunands to the slow effectors are given by steady-state inputs of the proportional-plus-integral type optimal regulator. To illustrate the effectiveness of the proposed methods, computer simulation is conducted using linear models of the B-747. Copyright © 1998 IFAC Keywords: Flight control, Propulsion control, H-infinity control, Equilibrium, Optimal regulators, Proportional plus integral action, Steady states

1. INTRODUCTION

15A in 1993. Following this, McDonnell Douglas and the center developed an autopilot for the MD-Il PCA and succeeded flight tests, including approach and landing (Burken and Burcham Jr. , 1997). Piloted simulation tests are also conducted for a B747-400 PCA by a group of NASA Ames Research Center (Bull, et. al., 1997).

Since hydraulic systems of large transport aircraft are redundant, it is rare that all the control surfaces of the aircraft become no use. However, several accidents caused by hydraulic failure really happened; for example, the accident of the B-74 7 of Japan Airlines in 1985 and that of the DC-lO of United Airlines in Sioux-City in 1989. Particularly, in the latter case the pilots controlled the aircraft using thrust only to manage to fly to the airport, saving 187 people out of the 298 crew and passengers. This example suggests the possibility of thrust control of aircraft. Since it is not easy for pilots to do this due to slow response of engines, etc., introducing automatic control to thrust control will alleviate pilots' workload and enhance safety (Burcham Jr., 1997).

The autopilots for the PCA are designed based on classical control. On the other hand, Jonckheere et. al. (1996) tried a state-space approach in autopilot design for the L-IOll PCA, which is model following control formulated as an H 00 control problem. Ochi and Kanai (1998) also proposed a model-following control system for the L-lOll PCA based on ~ state feedback control. However, it is more important for PCA to follow appropriate flight-path conunands than to recover desired or nominal aircraft dynamics by model following. In fact, whether a human pilot or autopilot gives control conunands to the model following flight control system (FCS), the whole

The aircraft controlled by thrust only is called Propulsion Controlled Aircraft (PCA). NASA Dryden Flight Research Center successfully conducted flight experiments of peA using the F-

l35

system including the (atuo)pilot in the outer loop may not function due to decrease of control power, unless the control gains are modified from the nominal ones. This paper proposes a flight -path control system for ILS(instrument landing system)-coupled approach. The FCS is designed by employing a modem control technique or H 00 state feedback method, which makes it possible to design the FCS at a time unlike step-by-step design methods, while guaranteeing internal stability. The proposed FCS is applied to a longitudinal linear model of the B-747 in approach configuration.

Glide Slope Transmitter (St ation)

Fig. I Geometry of the glide slope problem; Note d and rare negative. 2. DESIGN OF FLIGHT PATH CONTROLLER BASED ON IT' CONTROL 2.1 Geometry oflLS-coupled Approach Figure 1 shows a geometry of ILS-coupled approach (Blakelock, 1991). The desired flight path is given by a beam emitted from the glide slope transmitter. The control objective is to make the distance d between the aircraft and the beam or glide slope centerline (GSC) zero. Since d cannot be measured, the angle r in Fig. I is taken as an output to be controlled, in place of d. From Fig. I, r is expressed for a small d as

Hydraulic system failure sometimes causes control surface jam. Stuck control surfaces can generate significant disturbances, so that the failure may lead to a fatal accident, especially in large transport aircraft. Ochi and Kanai (1993) proposed a method to accommodate such failures by taking advantage of slow effectors such as stabilizers and engines. Command inputs to the slow effectors are step and detennined using a pseudo-inverse of the control derivative matrix in a linear aircraft model so that the force or moment produced by the stuck control surfaces can be counteracted by the slow effectors. This idea is the same as control mixer used in selfrepairing flight control (Huber and McCulloch, 1984). However, the method does not take account of effects given by the failure through aircraft dynamics. In fact, the system derivative matrix is not used in derivation of the control law. Therefore, the control law for the slow effector is not necessarily good, when the slow effectors cannot effectively counteract the force or moment generated by the stuck surfaces. This paper proposes a solution to this problem, which employs steady-state controls of the proportional-plus-integral (PI) type optimal regulator. The steady-state inputs are used as command inputs to the slow effectors, and the steady state variables are applied to modification of the trim point. Numerical simulation using a linear model of the B-74 7 demonstrates that the proposed control law and trim adjustment provides smaller deviation of the state variables in time responses for the failure of rudder jam.

r =~

(1)

Rn and d can be expressed as d

=

(2)

tVc(r + r"j)dr

where Vc is the speed with respect to the earth, r is the flight path angle, Yr,! is the angle between the GSC and the ground, and Rn is defined in Fig. 1. 2.2 FCS Design by H 00 State Feedback Control In H 00 control, controllers are detennined so that H 00 norm of the closed-loop transfer function from external inputs to specified criterion outputs can be minimized. Designing a state feedback controller by the H 00 control theory is called full information (PI) problem. Suppose that the aircraft dynamics are described by the linear time-invariant state equations and output equations: xp=Apxp+Bpu p

(3)

Yp-Cpxp+Dpu p

(4)

where xp=[u w q B 6;,,]T and up=b;"c; u is forward speed (m/s), w downward speed (m/s), q pitch rate (rad/s), B pitch angle (rad), 6;" engine thrust (N), and 6;"c thrust command to the engines (N). Defining YP as yp=[u -J1 T=[u _Ora]T, Cp and Dp are given as

In the following, first, an outline of the FCS design for PCA based on H00 control is shown. In Section 3, the failure-accommodation method using slow effectors is described, comparing it to the controlmixer approach. In Section 4, computer simulation results are shown. Finally, conclusions are given.

Cp -

[~ l/~o ~ ~1 ~] , Dp [~] =

where Vo is the nominal trim air speed. Introducing frequency weight to the cost function and taking account of Eq. (2) give dynamics to the controller. Let the dynamics be described by 136

x,

=

A,x, + B,u,

(5)

y,

=

c,x,

(6)

+ D,u,

where

where A c, Bc, Cc, and Dc are determined as

~=I-l~~o ~ ~} -1/~

and Dc -

°

3• 3 •

Be-r/;

0

0

~ ~} Ce-I~ ~ ~11 0

1

0 0

On.n denotes an nxn zero matrix.

D [D pTO J. p ..

Cp = [c /

or

Substituting Eq. (7) for

21

=

[Isxs 03.5

OSX3] , 03x3

D21 = 0S.3' and D22 - 08.1 ; I

...

denotes an nxn

identity matrix. Defining a state vector x as x=[x/ x/JT, the state equation can be written as

Uc is defined as control errors, i. e., u c=[ureru Yre~Y rreJ1T, where Urej and Yrej are references to U and 1, respectively. Defining Xc=[Xcl Xc2 Xc3]T, Xci is weighted air-speed error (u,.ru)/(T1s+ 1), Xc2 is d, and Xc3 is weighted control error (Ler O/s, where rrej is the reference to r , which is usually zero. Figure 2 shows a block diagram of the control system, including constant weights defined below. U c can be written as U, "" W, - C Px p - D pUp (7)

where wc=[urej Yrej Lej]T,

C

and Uc

(8)

Y e "" -D,C pXp + C,X, + D,w, - DeDpup

(9)

Thus, Eqs. (12), (10), and (11) describe a so-called generalized plant, whose block diagram is shown by Fig. 2. The optimal feedback control law is given

up~[KI

KJx/

x, Tf

(13)

where K=[KI K 2 ]=-B/X and X is the semi-positive definite solution of the following algebraic Riccati equation: X4 + ATX + X(YR-2BIB/ - B 2(D I/ D 12 )-IB/)X

in

Eqs. (5) and (6) yields the following equations x, - - B ,C p x p + A, x, + B, w, - B, D pUp

(12)

+CITCI -0 where ~ is a positive constant.

(14)

Choose the criterion output z as z=[alxcI a2xc3 Wub;hc]T or

In the controller design, note that RnO in Ac and V•. in Bc are treated as constants. Particularly, since Rn

Z=CIIXp+CI2X,+DlIW,+DI2Up

decreases, as the aircraft approaches the glide slop transmitter, loop gain of the control system increases so that stability margin decreases. Hence, Rn used in the design must be chosen carefully. By multiplying Xc3 by RjR no , this problem can be removed. However, since another measurement Rn becomes necessary, the gain-scheduling compensation is not taken in this paper.

where CII

=

°

3• 5 ,

DII = 03• 3 ,

I2[~I ~ :2]'

C =

(10)

°°°

and DI2 = [

~ l·

Wu The scalar weights aI , a2, and Wu are positive constants.

Uncertainties such as parameter variations or UTlffiodeled dynamics are not considered here.

Assuming that all the states of xp are measurable, the outputs to be fed back are y=[x/ xcT]T or

Z3

1 105 + 1

&hc

Aircraft & Engine Dynamics

xp

u -

+

1

Xci

1;5+1

Cp j+ Yref

K2 Fig. 2 Flight control system for ILS-coupled approach

137

4.11x10· 4 6.41x10· 2 O]TxO. l , which is quite small. The steady-state value of the roll angle is 9.47 deg; hence, the aileron-stuck failure is effectively accommodated by the stabilizers.

3. ACCOMMODATION OF CONTROL SURFACE JAM AND TRIM ADJUSTMENT

Breaking down the controls into three groups, rewrite Eq. (2) as xp = Apxp + Blul + B 2 u 2 + B 3 u 3 (15)

When the rudder is stuck at -0.1 rad, likewise u)=o,.=-O.l and B) is the second column of Bp; U2=0s1b and B2 is the third column of Bp; U3=O'a=0. In this case, u2=-O.0612 rad from Eq. (17), and the left-hand side of Eq. (18) is calculated as [2.91 0 0.0402 -0.0627 Ofx(-O.l), whose norm is larger than that in the case of aileron failure. In fact, applying U2 to the stabilizer, the roll angle increases and the aircraft cannot be trimmed. The reason for this is as follows : When the rudder is stuck at -0.1 rad, it produces negative rolling moment and positive yawing moment. Although the former can be counteracted by the stabilizers, the latter cannot; hence, the control law gives the stabilizer angle that counteracts the rolling moment. However, the yawing moment produces a large yawing motion that results in a large rolling motion by the difference of the lift between the right wing and the left wing. Besides, the direction of the rolling motion is opposite to the one directly generated by the rudder, which implies that the stabilizer angle helps the rolling motion by the yawing grow. Thus, the control mixer approach does not always work to accommodate control surface jam.

"I

where indicates deflection angles of stuck control surfaces, U2 control inputs of slow effectors, and "3 control inputs of other effectors, which are assumed to be null in this section.

3.1 Control Mixer (Method 1) This is an accommodation method that directly counteracts the force or moment produced by "I using "2; namely, U2 is determined so that the following equation can hold: (16) Bt"1 + B 2 "2 - 0 Equation (16) can be solved as u 2 = -Bz+Btut (17) where the superscript ' +' denotes pseudo-inverse. Since the number of rows of B2 is usually larger than the number of columns, the solution given by Eq. (17) is nothing more than an approximate one ofEq. (16). Particularly, when BI is orthogonal or nearly orthogonal to B 2 , preciseness of the approximation becomes bad, which means that "2 cannot accommodate the failure of "I . Numerical example: Let us show an example using a linear model of lateral-directional motion of the B747 at an altitude of 6,080 m and a speed of 205 m1s. The stability and control derivative matrices and the state and control vectors are, respectively, -0.104 8.94 -205 9.79 -0.0144 -0.804 0 .317 0 A p - 0 .00450 -0.0531 -0.193 0 1 0.0437 0 0 2.91 0 0

Bp

=

0.210

0.211

0 .279

0.0199

- 0.610

0.0179

0

o

0

3.2 Failure Accommodation using Steady-State

Control of the PI-type Optimal Regulator (Method 2) To remove the drawback of Method 1, it is necessary to take the aircraft dynamics or the stability derivative matrix into account. First, note that a stabilized system settles at an equilibrium point even in the presence of constant disturbances. Hence, "2 can be given by steady-state values of a stabilizing control law. Second, when "2 is determined in that way, it is wanted as small as possible. This can be achieved by imposing a frequency weight that increases in low frequencies, for example, s, on "2 in design of the stabilizing control law. Third, some specified states or outputs, for instance the roll rate in the above example, are wanted to damp fast and become specified values exactly. A controller that satisfies these requirements is the PI-type optimal regulator, and U2 can be given by its steady-state controls.

r,

x p - [v p r ~ and "p - [0'0 0' r 0' SIb ]T , where v is sideward speed (m1s), p roll rate (radls), r yaw rate (radls), ~ roll angle (rad), O'a aileron angle (rad), 0,. rudder angle (rad), and Oslb stabilizer angle (rad). The stabilizers are assumed to be able to move differentially.

Assuming that "3=0, Eq. (15) becomes xp = Apxp + Bt"1 + B 2 "2 (19) Defining control error, e=Yp-Yrej, and the cost function as

When the ailerons are stuck at 0.1 rad, then u)=O'a=O.l and B) is the first column of Bp; U2=0s1b and B2 is the third column of Bp; U3=O',=0 . Substituting Eq. (17) into Eq. (16) yields (/ - B2B2 ")B I " , - 0 (18) In this case, u2=-0.0753 rad from Eq. (17), and the left-hand side of Eq. (18) is calculated as [0 -

J '"

t

(eT Qe

+

u/ &2 )dt

(20)

the optimal control law that minimizes the cost function is determined as 138

"2 -K,rIXp +K"2 E (21) where Q and R are semi-positive and pOSItIve definite

respectively, E ~ fa'" edt ,

matrices,

and

[Ktrl K tr2 ]=-R -IB T p. P is a positive definite solution of the algebraic Riccati equation:

100

time (sec)

PA +ATp-PBR-IIFp+cTQc =0 where

A-

[~: ~],

B ..

[:2]. C .. [0

200

ac!S_1.------.-----r-------..--u 'U ¥ ____________ ~g>Or'" ~

C p ].

~~

~

··

-1 nI- - angle of attack - g>-2 - ----pitch angle °nl ~~ -3 \_-.------ ----------------------- ----ClU §:a. -4 OL--~---1....l...00--~--2-'OO time (sec)

·

I

The state variables [x/ .i]T in the steady state, which is denoted as [xps/ Cs/IT, are given by

[:~ 1~ _[A, ~~,K, B':r[~},

I I I

(22)

Substituting xpss and Css for xp and & in Eq. (21), respectively, the steady-state values of " 2 or the slow control effectors are detennined. And xpss gives variations of the trim states. Thus, the variation of the trim point is provided by the steady-state "2 and xpss. Since nonlinearity of the aircraft motion is not taken into account, the steady-state values do not give exact variation of the equilibrium point. However, unless the variation is large, they can be used to give a new trim point.

(X10j

o

Bp = [3 .913

o X

10-

-1.769

X

10 -'

is

Figure 3 shows time responses of some states. The forward speed is not controlled well; in fact, it increases by about 4 mls. The responses of angle of attack and pitch angle are reasonable. Note that the angles take negative values, but in practice they are positive, considering the trim angles, 5.7 deg. Figure 4 shows a flight path, including flare control for which another controller is used. [converges to zero so that the PCA is guided to the GSC without going out of the bearn-capture zone, which is If/sO.5 deg.

o

I 6 .996 x 10 -

8

WO

s, al=O.OI , a 2=1O, Wu=l , Yo=lOO. The reference inputs are u"rO mls, rrer2.5 deg, and ["rO deg. Initial conditions are given as xp=[O 0 0 0 O]T, x c=[O o O]T, and initial altitude ho=809 m. This means that the PCA in level flight captures the GSC 10 nm or 18520 m away from the GS transmitter. In computer simulation, although the aircraft model is linear, nonlinearity taken into account in kinematic and geometric relations.

4.1 Propulsion Control The aircraft model is given by longitudinal linear equations of motion of the B-747 in approach configuration in the flight condition of sea level and a speed of 84.5 mls. The trim angle of attack and pitch angle are 5.7 deg. The maximum cruise thrust is about 5.0x105 N. The matrices Ap and Bp of the equations of motion are given by -1.08CkI if 1.060<10-1 -8459 -9.766 1 8531 -1009 ~ _ -1.55lclcr -6.344< 103 -5.04~10-1 7.l6CkI~ 7.94Sdcr -5.83:>d06

100 time (sec)

Fig. 3 Time responses in ILS-coupled approach

4. SIMULATION

o

-:~'-------'---:,~:I

oT

Assuming that the trim thrust is half of the maximum one. the available thrust in the linear model is within :!:2.0x10 5 N. The engine dynamics, which is assumed to be described by a first-order delay system with a time constant of 1.5 s, are actually included in the aircraft model. The time constant is chosen to be more conservative than that used by Burken and Burcham (1997). Then the state variables are defined as in Section 2.2.

800

:[600

- - flight path - --- - - glide slope

~ 400

=

16200 o~-~--~--~--~---'

o

By trial an error, the following design parameters are chosen: R o =7625 In, Vc=Uo=84.5 mls, TI=O .OI

10000

horizontal range (m)

Fig. 4 Flight path in ILS-coupled approach and landing 139

REFERENCES

.Q.~600k2:1



Blakelock, J. H. (1991). Longitudinal Autopilots In: Automatic Control of Aircraft and Missiles. 2nd Ed., Chap. 2, John Wiley & Sons, Inc. , New York. Bull, J. , R. Mah, G. Hardy, B. Sullivan, J. Jones, D. Williarns, P. Soukup, J. Winters (1997). Piloted Simulation Tests of Propulsion Control as Backup to Loss of Primary Flight Controls for a B747-400 Jet Transport. NASA TM- 112191. Burcham Jr., F. W. (1997). Landing safely when flight controls fail. AL4A Aerospace America, October, 20-23 . Burken J. J. and F. W. Burcham Jr. (1997). FlightTest Results of Propulsion-Only Emergency Control System on MD 11 Airplane. AL4A Journal of Guidance. Control. and Dynamics, 20, 980-987. Huber, R. R. and McCulloch, B. (1984). SelfRepairing Flight Control System. SAE Technical Paper Series. 841552. Proc. of Aerospace Congress & Exposition, 1-20. Jonckheere, E. A. , Yu, G-R, and Chu, C-K(1996). Hoo Control of Crippled Aircraft with Throttles Only. Proc. of 13th IFAC World Congress, 219224. Ochi, Y. and Kanai, K. (1995). Application of Restructurable Flight Control System to Large Transport Aircraft. AL4A Journal of Guidance. Control. and Dynamics. 18, 365-370. Ochi, Y. and Kanai, K. (1998). Propulsion Control and Trim Adjustment for Impaired Large Transport Aircraft. Proc. of 2nd ICNPAA, Daytona Beach (to appear). Saeki, M., et. al . (1994). Control System Design, Chap. 4. Asakura, Tokyo (in Japanese).

n:::~ o

20

40 60 time (sec)

80

100

Fig. 5 Time responses by Method I

5 ~ CJ)Q) Q)-c

~CJ)CJ) O/S~ _ 0>

ei5

- - : roll angle ------: sideslip angle

0

-5 -10

0

20

40 60 time (sec)

80

100

Fig. 6 Time responses by Method 2

4.2 Trim Adjustment Using the aircraft model of the B-747 shown in Section 4, time responses are computed for the failure that the rudder is stuck at -5 .7 deg. The stabilizer angle obtained by Method I is 3.5 deg and that by Method 2 is -30.0 deg, where the weighting matrices are Q= I and R= 1. The rudder and stabilizer angles are given as step input and actuator dynamics are not taken into account. Time responses are shown in Figs. 5 and 6. From Fig. 5, it can be seen that using Method I , the roll angle grows very large, which implies that the disturbances caused by the rudder are not rejected effectively. By contrast, as Fig. 6 shows, the stabilizer angle given by Method 2 can reject the disturbances. The state vector in the steady state is 2 X ss= [-3 .24 0 0 _9 .35xlO- ]T, which approximately gives variation of the trim states.

5. CONCLUSIONS A flight control system for PCA is designed based on H 00 state feedback control. The FCS can achieve ILS-coupled approach for the B-74 7 using thrust only. Meanwhile, this paper proposes a method that accommodates control surface jam using slow effectors. It is shown through computer simulation that the method based on the PI-type optimal regulator is much more effective than the controlmixer method, especially in the case where airframe dynamics are involved. A problem with this method is how to give weighting matrices of the cost function.

140