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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS:Vol. 70, No. 1, JULY !991

Aircraft Control for Flight in an Uncertain Environment: Takeoff in Windshear G. LEITMANN 1 AND S. PANDEY 2

Abstract. The design of the control of an aircraft encountering windshear after takeoff is treated as a problem of stabilizing the climb rate about a desired value of the climb rate. The resulting controller is a feedback one utilizing only climb rate information. Its robustness vis-avis windshear structure and intensity is illustrated via simulations employing four different windshear models. Key Words. Aircraft guidance, takeoff in windshear, control of uncertain systems.

Notations A R L & aircraft reference line; D & drag force, lb; g-~ gravitational force per unit mass = const, ft sec -2; h &vertical coordinate of aircraft center of mass (altitude), ft; L g lift force, lb; m & aircraft mass = const, lb f t - 1 sec2; O ~ mass center of aircraft; S&reference surface, ft2; t & time, sec; T g thrust force, lb; V-4 aircraft speed relative to wind-based reference frame, ft sec- 1; Ve& aircraft speed relative to ground, ft sec- 1; Wx ~ horizontal component o f wind velocity, ft sec- 1; Wh ~ vertical component of wind velocity, ft sec- 1; x g horizontal coordinate of aircraft center of mass, ft;

1Professor, College of Engineering, University o f California, Berkeley, California. 2Graduate Student, Department o f Mechanical Engineering, University of California, Berkeley, California.

25 0022-3239/91/0700-0025506,50/0 © 1991PlenumPublishingCorporation

26

JOTA: VOL. 70, NO. 1, JULY 1991

a&relative angle of attack, rad; y--arelative path inclination, rad; ~e~ path inclination, rad; ~-~ thrust inclination, rad; p =~air density = const, lb ft 2 sec 2; Dot denotes time derivative.

1. Introduction

The problem of guiding an aircraft encountering windshear has gained considerable importance since a 1977 FAA study revealed low-level windshear as a contributing factor in many accidents involving large aircraft; see, e.g., Refs. 1 and 2. Much effort has gone into modeling and identifying windshear; see, e.g., Refs. 3-6. Other investigations have been concerned with the design of controllers to enhance the chances for survival while encountering windshear during takeoff or landing. Primary among these have been the pioneering studies of Miele; see, e.g., Refs. 2, 7, 8, 9 and many others referenced in Ref. 10. The so-called simplified gamma guidance scheme, espoused by Miele, is one based on attaining near-optimal trajectories in the presence of a given windshear structure; it has been shown to have good survival capability in the prescribed windshear model. Another major contributor to the topic of aircraft control is Bryson; see, e.g., Ref. 11, where the guidance scheme consists of a nonlinear nominal control together with a linear feedback designed to stabilize the response of a linearized system about a desired nominal trajectory. Yet another approach has been via deterministic control of uncertain systems3; see, e.g., Refs. 12-15. While Refs. 12 and 13 employ angle-of-attack control, the former to stabilize all state variables and the latter only the relative path inclination, Ref. 14 utilizes the difference between angle of attack and its nominal value as control to stabilize relative path inclination deviation from its desired value; this results in a considerably smoother angle of attack history. Finally, the Soviet efforts, exemplified by Ref. 15, employ a game-against-nature approach to deal with the uncertain environment due to windshear. These methods of control design for uncertain systems do not utilize any a priori information about the system uncertainties; they do require a priori assumptions on uncertainty bounds, here bounds of wind and wind rate of change.

3For a survey of such methods, see Ref. 16 and Appendix B.


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In this paper, we present yet another approach to aircraft guidance in windshear. In particular, we consider the control of the vertical velocity component (rate of climb) by means of the angle-of-attack deviation from its nominal value. In counter-distinction to Refs. 12-15, no apriori bounding information is needed.

2. Aircraft Equations of Motion Following Miele's lead, we employ equations of motion for the center of mass of the aircraft, in which the kinematic variables are relative to the ground (inertial reference frame), while the dynamic ones are taken relative to a moving but nonrotating reference frame translating with the wind velocity at the aircraft center of mass (wind-based reference frame).

Assumptions. (A1) (A2) (A3) (A4) (A5)

The rotational inertia of the aircraft and the sensor and actuator dynamics are neglected. The aircraft mass is constant. Air density is constant. Flight is in the vertical plane. Maximum thrust is used.

Assumption A2 presupposes that the amount of fuel used during the period under consideration is small compared to the total aircraft mass. Assumption A3 is well founded, since the flight regime of concern spans a small altitude change (about 1500 ft). Assumptions A4 is justified for takeoff (but not for landing, in general). Assumption A5 corresponds to the usual practice of using full throttle during takeoff. In view of Assumption A1, we consider only the equations of motion of the center of mass (see Fig. 1). The kinematical equations 4 are ~ = Vcos 7+ Wx,

(1)

/~= Vsin 7+ Wh,

(2)

and the dynamical equations are rnI~= Tcos(a+ ~ ) - D - m g sin 7-m(I/Vx cos 7+ l~h sin 7),

(3)

mV~'=Tsin(a+~)+L-mg c°s 7+m(li/x sin 7 - l~h COS 7)-

(4)

4For the sake of brevity, we shall delete the arguments of functions whenever this does not entail loss of clarity.

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ARL

Ve L

T

I m

~

a

D

mg

Fig. 1. Coordinate system and free body diagram.

These equations must be supplemented by specifying the thrust force

T= T(V), the drag D=D(V, a), the lift L=L(V, a), the horizontal wind Wx = Wx(x, h) or Wx(t), and the vertical wind Wh = Wh(x, h) or Wh(t). For a given value of the thrust inclination ~, the differential equation system (1)(4) involves four state variables [the horizontal distance x(t), the altitude h(t), the relative speed V(t), and the relative path inclination 7/(0] and one control variable [the angle of attack a(t)], since maximum thrust is employed according to Assumption A4. 2.1. Bounded Quantities. In order to account for the aircraft capabilities, we shall assume that there is a maximum attainable value of the relative angle of attack a, that is, a~ [0, a,],

(5)

where a, > 0 depends on the specific aircraft and generally is taken to be the stick-shaker angle of attack. To account for the neglected dynamics of rotation, as well as of sensors and actuators, we bound the attainable magnitude of the rate of change of the relative angle of attack ~, that is,

lat-< C, where C> 0 depends on the specific aircraft.

(6)


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Furthermore, the range of practical values of the relative aircraft speed V is limited, that is,

y_< v_< ~',

(7)

where_V> 0 and ~'>V depend on the specific aircraft. These bounds correspond, for instance, to the relative stall speed and the maximum attainable relative speed, respectively. The bounds (5) and (6) on a and tel will be neglected in deducing the proposed aircraft guidance scheme; however, they will be taken into account in the numerical simulations. On the other hand, the bounds (7) on the relative speed V will be employed in the construction of the proposed guidance scheme. 2.2. Approximations for the Force Terms.

Thrust.

The thrust T is approximated as

T= Ao + A 1V+ A2 V 2,

(8)

where the coefficients Ao, A~, A2 depend on the altitude of the runway, the ambient temperature, and the engine power setting. Drag.

The drag D is written in the form

D = (1/2)CDpSV 2,

(9a)

CD=Bo+ B~a + Bza 2.

(9b)

where

The coefficients Bo, B~, B2 depend on the flap setting and the undercarriage position. Lift.

The lift L is written as L = (1/2)CLpSV 2,

(10)

where Cr = Co + C1a,

if aN a** < a.,

(11)

CL=Co+C~a+Cz(a-a**) 2,

if a** to. (ii) Uniform Boundedness. Given re(0, ~ ) , there exists a positive d(r) < ~ such that, for all solutions y(- ):[to, tl)~N n, y(to) =Yo, of (33), ]byol[ d , there is a positive ~(~ such that, for every solution y(. ):[to, ~ ) ~ E ~ , y(to) =Y0, of (33),

yo~Gr(z~y(t)~G~,

for all t> to.

9. Appendix B: Stabilizing Control The following theorem (see Ref. 16 and references therein) is useful in the construction of practically stabilizing sets of controllers. Theorem 9.1. Consider an uncertain system described by (31) with oge~, and suppose that P is a collection of feedback control functions p: ~ x N"---,R m. If there exists a candidate Lyapunov function V: ~ x N n___,~ ÷ and a class K function or: R + ~ ~ ÷ such that, for each e > 0, there exists p ' e P which assures that, for all o9s ~,

~(t) = F(t, y(t), P'(4 y(t)), co)

(34)

has existence and indefinite extension of solutions and ~V

~V

,

( t, y) + ~y ( t, y)F( t, y, p ( t, y), o9)