I I

. AERO-ASTRONAUTICS REPORT NO. 203

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FINAL REPORT ON NASA GRANT NO. NAG-1-516, -- ---+ -

OPTIMAL FLIGHT TRAJECTORIES IN THE PRESENCE OF WINDSHEAR, 1984-86

A. MIELE

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(NASA-CR-1803 16) OPTIIlALL P L I G B I 887-26047 lBAJECTOBIES IN TEE P R E S E N C E CE UlblDSttBAE, 1984-86 F i n a l fiepcrt (Rice C e i v , ) 36 p A v a i l : l T X S HC A03/8€ A01 CSCL O l C Unclas

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RICE UNIVERSITY

1986

https://ntrs.nasa.gov/search.jsp?R=19870016614 2018-06-01T22:48:24+00:00Z

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AERO-ASTRONAUT1 CS REPORT NO. 203

FINAL REPORT ON NASA GRANT NO. NAG-1-516,

OPTIMAL FLIGHT TRAJECTORIES

I N THE'PRESENCE OF WINDSHEAR,'1984-86

bY

A. MIELE

RICE UNIVERSITY

1986

i AAR-203

Final Report on NASA Grant No. NAG-1-516,

Optimal F1 i g h t Trajectories 1 i n the Presence o f Windshear, 1984-86

bY 2 A. Miele

’ T h i s work was supported by NASA-Langley Research Center, Grant No. NAG-1-51 6.

‘Professor of Astronautics and Mathematical Sciences , Aero-Astronautics Group, Rice University, Houston, Texas.

i i AAR-203

Abstract. T h i s report summarizes the research performed a t Rice University

d u r i n g the period 1984-86 under NASA Grant No. NAG-1-516 on optimal f l i g h t

t ra jec tor ies i n the presence of windshear. With particular reference to the

take-off problem, the topics covered include: equations of motion, problem

formulation, algori thms, optimal f l i g h t t ra jector ies , advanced guidance schemes,

simplified guidance schemes, and p i l o t i n g strategies.

Key Words. Flight mechanics, take-off, optimal t ra jec tor ies , optimal control,

feedback control, windshear problems, sequential gradient-restoration algorithm,

dual sequential gradient-restoration algorithm, guidance s t ra tegies , acceleration

guidance, g a m guidance, theta guidance, piloting s t ra tegies .

iii AAR-203

Con tents

1. In t roduc t ion

2. Research Review

3. Pub1 i c a t i o n s

4. Abstracts of Publ icat ions

5. Bib1 iography

Page 1

Page 2

Page 10

Page 13

Page 26

Acknowl edgment

The co l labora t ion o f Dr. T. Wang, Senior Research Associate, Rice Univers i ty ,

Houston, Texas i s g r a t e f u l l y acknowledged.

co l labora t ion o f Captain W. W. Melvin, Delta A i r l i n e s , At lanta, Georgia.

Melvin i s Chairman o f the Airworthiness and Performance Comnittee o f the A i r

L ine P i l o t s Associat ion (ALPA), Washington, DC. He introduced the author o f

t h i s r e p o r t t o the windshear problem and acted as a consul tant on t h i s p ro jec t .

Also g r a t e f u l l y acknowledged i s the

Captain

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1. Introduction

The objective of this s tudy i s the determination of optimal f l i g h t

t ra jec tor ies i n the presence of windshear and the development of guidance

schemes fo r near-optimum f l i g h t i n a windshear.

by sharp change i n intensity and direction over a re la t ively small region of

space.

airplanes and mili tary airplanes and i s key t o a i r c r a f t safety.

T h i s i s a wind characterized

T h i s problem is important i n the take-off and landing of both c iv i l ian I

The research done under NASA Grant No. NAG-1-516 d u r i n g the period 1984-86

is reviewed i n Section 2.

the topics covered include: equations of motion, problem formulation, algorithms,

optimal f l i g h t t ra jector ies , advanced guidance schemes, simplified guidance

schemes, and piloting strategies.

With particular reference t o the take-off problem,

The publications completed or i n progress a r e l isted i n Section 3. The

abstracts o f the publications completed are given i n Section 4.

bibliography is presented i n Section 5.

Finally, a

2 AAR-203

2. Research Review

2.1. Background. During the month of November 1983, Captain GJ. W. Melvin,

Del ta A i r l i n e s and ALPA, approached D r . A. Miele, Professor o f Astronautics

and Mathematical Sciences a t Rice University, asking him t o become in te res ted

i n the windshear problem. I n a meeting which took place on the campus o f Rice

Universi ty, Captain Melvin stated h i s fee l i ng t h a t considerable research had

been done on the meteorological, aerodynamic, instrumentation, and s t a b i l i t y

aspects o f the windshear problem; however, r e l a t i v e l y l i t t l e had been done on

the f l i g h t mechanics aspects; he f e l t t ha t a fundamental study was needed i n

order t o b e t t e r understand the dynamic behavior o f an a i r c r a f t i n a windshear.

I n the ensuing discussion, Dr. Miele stated h i s f e e l i n g t h a t the determination

o f good s t ra teg ies f o r coping wi th windshear s i t u a t i o n s was e s s e n t i a l l y an

optimal con t ro l problem; t h a t the methods o f optimal con t ro l theory were needed;

and that, on l y a f t e r having found optimal con t ro l solut ions, one could proper ly

address the guidance problem.

As a r e s u l t o f t h i s meeting, Dr . Miele prepared a research proposal

on the opt imizat ion and guidance o f f l i g h t t r a j e c t o r i e s i n a windshear. The

proposal was funded i n August 1984 by NASA-Langley Research Center, w i t h

Dr. Miele ac t i ng as P r inc ipa l Invest igator and Captain Melvin ac t i ng as

Consultant. D r . R. L. Bowles o f NASA-LRC i s P ro jec t Monitor. Addi t ional

funds were subsequently obtained through the sponsorship o f Boeing Comerc ia l

A i r c r a f t Company (BCAC) and A i r L ine P i l o t s Associat ion (ALPA).

During the past two years, the Aero-Astronautics Group o f Rice Universi ty,

under the d i r e c t i o n o f Dr . Miele, has done research on both the op t im iza t i on

and the guidance of f l i g h t t r a j e c t o r i e s i n a windshear.

the Aero-Astronautics Group studied the opt imizat ion aspects of f l i g h t

t r a j e c t o r i e s (Refs. 1-7 and 14); i n the second year , the Rero-Astronautics

I n the f i r s t year,

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Group studied the guidance aspects (Refs, 8-13 and 15-19).

are summari zed bel ow.

The main resul ts

2.2. Equations of Motion. In Ref. 1 , the equations of motion under

1 windshear conditions are derived employing three different coordinate systems:

the Earth-fixed system; the relat ive wind-axes system; and the absolute

wind-axes system. The following assumptions are employed: the a i r c r a f t i s a

par t ic le of constant mass; f l i g h t takes place i n a vertical plane; Newton's

law i s valid in an Earth-fixed system; and the wind flow f i e ld is steady.

For the optimization of f l i g h t t ra jector ies , any of the previous

coordinate systems can be used. However, the re la t ive wind-axes system i s t o

be preferred, because the windshear terms appear explicity i n the dynamical

equations.

physical understanding and interpretation of windshear phenomena.

Therefore, use of the relative wind-axes system allows an easier

2.3. Problem Formulation. In Ref. 2, we employ the equations of

motion written fo r the re la t ive wind-axes system. First, we supply an analytical

description of the forces acting on the aircraf t . Next, we supply a description

of the wind flow f ie ld .

however, useful one-dimensional-models can be developed i f one refers t o the

near-the-ground behavior of a microburst.

Generally speaking, the wind flow f i e ld i s two-dimensionai;

W i t h reference t o take-off, we assume t h a t the power set t ing i s held a t

the maximum value. Indeed, i t i s logical t o t h i n k tha t , i f a plane takes off

under less-than-ideal weather conditions, a prudent p i lo t employs the maximum

thrust.

only control i s the angle of attack a.

inequality constraints must be imposed on bo th c1 and &.

of attack a i s subject t o the inequality a - < a*, where a, i s a prescribed upper

bound.

W i t h the power set t ing held a t the maximum value, i t i s c lear tha t the

To obtain r e a l i s t i c t ra jec tor ies ,

Specifically, the angle

I n a d d i t i o n , i t s time derivative & i s subject t o the double inequality

4 AAR-203

-C - - < & < +C, where C i s a prescribed constant.

Concerning the i n i t i a l conditions, we assume tha t the i n i t i a l s t a t e

i s given.

four types of boundary conditions a re considered.

Concerning the final conditions, four cases a re considered; hence,

( B C O ) The s t a t e is f r ee a t the final point.

(BC1) The final value of the pa th inclination i s the same as the

i n i t i a l value.

( B C Z ) The final values of the velocity and the path inclination a re the

same as the i n i t i a l values.

(BC3) The final values of the velocity, the path inclination, and the

angle of attack a r e the same as the in i t ia l values. Therefore, this case implies

that , i f the i n i t i a l values correspond t o quasi-steady f l i g h t , t h e n the f ina l

values a lso correspond to quasi-steady f l ight . I I

I

Concerning the performance indexes, we consider e i g h t fundamental

opt i m i za ti on pro b 1 ems.

( P l ) . T h i s i s a least-square problem involving Ah = h - hR, the difference

between the f l i g h t a l t i t ude and a reference a l t i t ude , assumed t o be a l inear

function of the horizontal distance.

(P2) T h i s i s a least-square problem involving Ay = y - yR, the

difference between the re la t ive path inclination and a reference value, assumed

constant.

(P3) T h i s i s a least-square problem involving Aye = ye - yeR, the

difference between the absolute path inclination and a reference value, assumed

constant.

(P4 ) This i s a minimax problem involving Ah = h - h R , the difference

between the f l i g h t a1 t i t ude and a reference a1 t i t u d e , assumed constant.

(P5) This i s a minimax problem involving G A h = G ( h - h R ) , the weighted

difference between the f l i g h t a l t i t ude and a reference a l t i tude , assumed constant,

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as i n (P4). Here, G( t ) i s a prescribed weighting funct ion.

hR, the

tude, assumed t o be

(P6) This i s a minimax problem involv ing Ah =

difference between the f l i g h t a1 ti tude and a reference

a l i n e a r func t i on o f the hor izontal distance, as i n (P

This i s a minimax problem involv ing Ay = (P7) the d i f f e rence YR between the r e l a t i v e path i n c l i n a t i o n and a reference value, assumed constant,

as i n (P2).

(P8). This i s a minimax problem invo lv ing Aye = ye - yeR, the

d i f ference between the absolute path i n c l i n a t i o n and a reference value, assumed

constant, as i n (P3).

Probl ems (P1 ) -( P3) a re 1 east-square probl ems o f the Bo1 za type. Probl ems I

(P4)-(P8) a re minimax problems o f the Chebyshev type, which can be converted

i n t o Bolza problems through su i tab le transformations.

are p a r t i c u l a r cases o f t he fo l lowing general problem:

Hence, Problems ( P l )-(P8)

(P) Minimize a funct ional w i t h respect t o the s t a t e vector x ( t ) , the

contro l vector u( t ) , and the parameter vector IT which s a t i s f y a system of

d i f f e r e n t i a l constraints, i n i t i a l constraints, and f i n a i const ra in ts .

2.4. Algorithms. From the previous section, i t i s c l e a r t h a t one i s faced

w i t h a wide v a r i e t y o f problems o f optimal cont ro l , depending on the p a r t i c u l a r

performance index chosen and the p a r t i c u l a r type o f boundary condi t ions chosen.

These problems are f u r t h e r complicated by the presence o f i n e q u a l i t y const ra in ts

on the contro l ( a ) and the t ime der ivat ive o f the con t ro l (4). Therefore, a

powerful a lgor i thm i s necessary t o solve the problems under consideration.

I n Ref. 3, we present the algori thm useful f o r so l v ing Problem (P)

on a d i g i t a l computer, more s p e c i f i c a l l y , the sequential gradient - restorat ion

algor i thm (SGRA).

presented.

Both the primal formulation and the dual formulat ion are

Depending on whether the primal formulation i s used o r the dual

h -

a1 t

1- Y -

6 AAR-203

formulation i s used, one obtains a primal sequential gradient-restoration

algorithm (PSGRA) o r a dual sequential gradient-restoration algorithm (DSGRA) . The systems of Lagrange mu1 t i p l i e r s associated w i t h the gradient phase

of SGRA and the restoration phase of SGRA a r e examined.

i t is shown tha t the Lagrange multipliers a r e endowed w i t h a dual i ty property:

they minimize a special functional , quadratic i n the mu1 t i p l iers, subject to

the mu1 t i p l i e r different ia l equations and boundary conditions, f o r given s t a t e ,

control , and parameter. These dual i ty properties have considerable computational

implications: they allow one t o reduce the auxiliary optimal control problems

associated w i t h the gradient phase and the restoration phase of SGRA t o

mathematical programming problems invo ving a f inite number of parameters as

unknowns.

For each phase,

Numerical experimentation has shown t h a t , f o r nonstiff problems of f l i g h t

mechanics, DSGRA is computationally more e f f i c i en t than PSGRA.

fo r the problems under consideration, DSGRA is about 10% more e f f i c i en t tha t

PSGRA. Hence, the subsequent numerical experiments a re based on the use of

DSGRA.

In par t icular ,

2.5. Optimal Trajectories. Optimal t ra jec tor ies were computed f o r the

Boeing B-727 a i r c r a f t , u s i n g the sequential gradient-restoration algorithm and

the NAS-AS-9000 computer of Rice University. Among the performance indexes

( P l ) - ( P 8 ) , the most re l iab le one was found to be (P8), based on the deviation

of the absolute path inclination from a reference value. After computing several

hundred optimal t ra jec tor ies , certain general conclusions became apparent (see

Refs. 1-7 and 14) :

( i ) the optimal t ra jector ies achieve minimum velocity near the time

when the shear ends;

7

~~

AAR-203

( i i ) the optimal t ra jector ies require an i n i t i a l decrease i n the angle

of attack, followed by a gradual increase; the maximum permissible angle of

attack OL* (stick-shaker angle of a t tack) is achieved near the time when the shear

ends;

( i i i ) for weak-to-moderate windshears,the optimal t ra jec tor ies a re

characterized by a monotonic climb; the average value of the path inclination

decreases a s the intensity o f the shear increases;

( i v ) fo r re la t ively severe windshears, the optimal t ra jec tor ies a r e

characterized by an i n i t i a l climb, followed by nearly horizontal f l i g h t ,

followed by renewed climbing a f t e r the a i r c ra f t has passed through the shear

region ;

( v ) weak-to-moderate windshears and re1 at ively severe w i ndshears a re

survivable employing an optimized f l i g h t strategy; however, extremely severe

windshears a r e not survivable, even employing an optimized f l i g h t strategy;

( v i ) i n re la t ively severe windshears, optimal t ra jec tor ies have a

much better survival capabili ty than constant angle of a t tack t ra jec tor ies

( fo r instance, maximum angle of attack t ra jector ies o r maximum 1 ift-to-drag

r a t i o t ra jec tor ies ) ; i n addition, they have a bet ter survival capabili ty

than constant pitch t ra jector ies .

2.6. Advanced Guidance Schemes. T h e computation of the optimal t ra jec tor ies

requires global information of the wind flow f i e ld ; t ha t i s , i t requires the

knowledge of the wind velocity components a t every point of the region of space

i n which the a i r c r a f t i s flying. In practice, this global information i s not

available; even i f i t were available, there would not be enough computing

capabili ty onboard and enough time t o process i t adequately. As a consequence,

one must t h i n k o f optimal t ra jec tor ies as merely benchmark t ra jec tor ies tha t i t

i s desirable t o approach i n actual f l i g h t .

8 AAR-203

Since global information i s not available, the best that one can do is

t o employ local information on the wind flow f i e ld , i n particular, local

information on the wind acceleration and the downdraft. Therefore, the

guidance problem must be addressed i n these terms: Assuming tha t local information

i s available on the wind acceleration, the downdraft, as well as the s t a t e

of the a i r c r a f t , we w i s h t o guide an a i r c ra f t automatically or semiautomatically

i n such a way tha t the key properties of the optimal t ra jec tor ies are preserved.

Based on the idea of preserving the properties of the optimal t ra jec tor ies ,

four guidance schemes were developed a t Rice University:

( a )

( b )

(c)

( d )

The de ta i l s of these guidance schemes a re omitted and can be found i n

acceleration guidance, based on the r e l a t ive acceleration;

absolute gama guidance, based on the absolute path inclination;

re la t ive gama guidance, based on the relat ive path incliantion;

theta guidance, based on the p i t c h a t t i t u d e angle.

Refs. 8-13 as well as i n Refs. 15-17.

Guidance t ra jector ies were computed f o r the Boeing B-727 a i r c r a f t us ing

I t the above guidance schemes and the HAS-AS-9000 computer of Rice University.

was found tha t a l l of the above guidance schemes ( i n particular, the acceleratian

guidance and the gama guidance) produce t ra jec tor ies which -are quite close t o

the optimum trajector ies .

a re superior to the t ra jector ies arising from al ternat ive guidance schemes

( for instance, constant angle of attack t ra jec tor ies and constant pitch t ra jec tor ies ) .

In addition, the result ing near-optimum t ra jec tor ies

2.7. Simp1 if ied Guidance Schemes and P i l o t i n g Strategies. As stated

above, the previous advanced guidance schemes require local information on the

wind flow f ie ld and the s t a t e of the aircraf t . While this information will be

available i n future a i r c ra f t , i t m i g h t n o t be available on current a i r c ra f t .

9 AAR-203

For current a i r c ra f t , one way t o survive a windshear encounter is to

use a constant pitch a t t i t ude technique; t h i s technique has been advocated by

many experts on the windshear problem.

is the quick t ransi t ion to horizontal f l i g h t , based on the properties o f the

optimal t ra jector ies .

An a l ternat ive, simple technique

T h e quick t ransi t ion t o horizontal f l i g h t requires an i n i t i a l decrease

of the angle of attack, so as t o decrease the path inclination to nearly

horizontal. Then, nearly horizontal f l i g h t is maintained d u r i n g the shear

encounter.

For re la t ively severe windshears, the quick horizontal f l i g h t t ransi t ion

technique yields t ra jec tor ies which a r e competitive w i t h those of the advanced

guidance schemes discussed previously. I n addition, f o r re la t ive ly severe

windshears, the quick horizontal f l i g h t transit ion technique yields t ra jec tor ies

which have better survival capabi l i t ies than those associated w i t h other guidance

techniques, such as constant angle o f attack o r constant pitch.

Uork on the quick horizontal f l igh t t ransi t ion technique is nearly

completed and publication of two reports is expected short ly (Refs. 18-19).

3.

3.1.

3.2.

3.3.

3.4.

3.5.

3.6.

3.7.

3.8.

Pub1 i c a t i o n s

MIELE, A., WANG, T., and MELVIN, W. W., Optimal F l i g h t T ra jec to r ies i n

the Presence o f Windshear, P a r t 1, Equations o f Motion, Rice Univers i ty ,

Aero-Astronautics Report No. 191, 1985.

MIELE, A., WANG, T., and MELVIN, W. W., Optimal F l i g h t T ra jec to r ies i n

the Presence o f Windshear, P a r t 2, Problem Formulation, Take-Off, Rice

Universi ty, Aero-Astronautics Report No. 192, 1985.

MIELE, A., WANG, T., and MELVIN, W. W., Optimal F l i g h t T ra jec to r ies i n

the Presence o f Windshear, Pa r t 3, Algorithms, Rice Universi ty,

Aero-Astronautics Report No. 193, 1985.

,

MIELE, A., WANG, T., and MELVIN, W. W., Optimal F l i g h t T ra jec to r ies i n

the Presence o f Windshear, P a r t 4, Numerical Results, Take-Off, Rice

Univers i ty , Aero-Astronautics Report No. 194, 1985.

MIELE, A., Sumnary Report on NASA Grant No. NAG-1-516, Optimal F l i g h t

T ra jec to r ies i n the Presence o f Windshear, 1985-86, Rice Univers i ty ,

Aero-Astronautics Report No. 7 95, 1985.

MIELE, A., WANG, T., and MELVIN, W. W., Optimal Take-Off T ra jec to r ies i n

the Presence of Windshear, Paper No. AIAA-85-1843-CP, A IAA F l i g h t Mechanics

Conference, Snowmass, Colorado, 1985.

MIELE, A., WANG, T., and MELVIN, W. W . , Optimal Take-Off T r a j e c t o r i e s

i n the Presence o f Windshear, Journal o f Opt imizat ion Theory and

Appl icat ions, Vol. 49, No. 1, pp. 1-45, 1986.

MIELE, A., WANG, T., and MELVIN, \I. W., Guidance Strateg ies f o r Near-

Optimum Performance i n a Windshear, P a r t 1, Take-Off, Basic Strategies,

Rice Universi ty, Aero-Astronautics Report No. 201, 1986.

11 AAR-203

3.9. MIELE, A., WANG, T., and MELVIN, W. H., Guidance Strategies for Near-

Optimum Performance i n a Windshear, Part 2, Take-Off, Comparison Strategies,

Rice University, Aero-Astronautics Report No. 202, 1986.

3.10. MIELE, A., WANG, T., and MELVIN, W. W., Guidance Strategies for Near-

Opt imum Take-Off Performance i n a Windshear, Paper No. AIAA-86-0181,

AIAA 24th Aerospace Sciences Meeting, Reno, Nevada, 1986.

MIELE, A., WANG, T., and MELVIN, W. W., Guidance Strategies f o r Near-

Optimum Take-Off Performance i n a Windshear, Journal of Optimization

Theory and Applications, Vol. 50, No. 1 , pp. 1-47, 1986.

3.11.

3.12. MIELE, A., WANG, T., and MELVIN, W. W., Optimization and Acceleration

Guidance of F l i g h t Trajectories in a Windshear, Paper No. AIAA-86-2036-CP,

AIAA Guidance, Navigation, and Control Conference, Will iamsburg, Virginia,

1986.

3.13. MIELE, A., WANG, T., and MELVIN, W. W., Optimization and Gama/Theta

Guidance of F l i g h t Trajectories in a Windshear, Paper No. ICAS-86-564,

15th Congress of the International Council o f the Aeronautical Sciences,

London, England, 1986.

MIELE, A., WANG, T., and MELVIN, W. W., Quasi-Steady Flight t o Quasi-

Steady F1 i g h t Transition i n a Windshear: Trajectory Optimization, 6th IFAC

Workshop on Control Applications of Nonlinear Programming and Optimization,

3.14.

London, England, 1986.

12 AAR- 203

Publications i n Progress

MIELE, A., e t a l , Optimization and Acceleration Guidance of F l i g h t

Trajectories in a Windshear, Journal of Guidance, Control, and Dynamics

( t o appear).

MIELE, A., e t a l , Downdraft Effects on Optimization and Guidance of

Take-Off Trajectories i n a Windshear, Journal of Optimization Theory

and Applications ( t o appear).

MIELE, A., e t a l , guasi-Steady Flight to Quasi-Steady F l i g h t Transition

i n a Windshear: Trajectory Guidance, Paper No. AIAA-87-0271, AIAA

25th Aerospace Sciences Meeti ng , Reno, Nevada, 1987 ( to appear).

MIELE, A., e t a l , Near-Optimal Piloting Strategies fo r F l i g h t i n Severe

Wi ndshear ( t o appear).

MIELE, A., e t a l , Simple P i l o t i n g Strategies f o r F l i g h t i n Severe

Windshear: Q u i c k Transition t o Horizontal Flight ( to appear).

3.15.

3.16.

3.17.

3.18.

3.19.

13 AAR-203

4. Abstracts of Pub1 ications

4.1. MIELE, A., WANG, T., and MELVIN, W. W., Optimal F l i g h t Trajectories i n

the Presence of Windshear, Part 1 , Equations of Motion, Rice University,

Aero-Astronautics Report No. 191, 1985.

Abstract. This report is the f i r s t of ser ies dealing w i t h the determination

of optimal f l i gh t t ra jec tor ies i n the presence of windshear.

characterized by sharp change i n intensity and direction over a re la t ive ly small

region of space. T h i s problem is important i n the take-off and landing of

both c iv i l ian airplanes and mili tary airplanes and i s key to a i r c r a f t safety.

T h i s is a wind

I t i s assumed that: the a i r c r a f t i s a par t ic le of constant mass; f l i g h t

takes place i n a vertical plane; Newton's law is valid i n an Earth-fixed system;

and the wind flow f ie ld is steady.

Under the above assumptions, the equations of motion under windshear

conditions a r e derived employing three different coordinate systems: the

Earth-fixed system; the re la t ive wind-axes system; and the absolute wind-axes

system. Transformations a re supp l i ed which allow one t o pass from one system

to another.

4.2. MIELE, A., WANG, T., and MELVIN, W. W., Optimal F l i g h t Trajectories i n

the Presence of Nindshear, Part 2, Problem Formulation, Take-Off, Rice

University, Aero-Astronautics Report No. 192, 1985.

Abstract. This report i s the second of a se r ies dealing w i t h the

determination of optimal f l i g h t t ra jector ies i n the presence of windshear.

employ the equations of motion written for the re la t ive wind-axes system. We

supply an analytical description of the forces acting on the a i r c r a f t , as well

as a description of the wind flow f i e ld .

we formulate eight fundamental optimization problems [Problems ( P l ) - ( P 8 ) 1

We

T h e n , w i t h reference to take-off,

under the assumptionsthat the power sett ing i s held a t the maximum value and

14 AAR-203

tha t the airplane is controlled through the angle of attack.

Problems (Pl)-(P3) a re least-square problems of the Bolza type. Problems

( P 4 ) - ( P 8 ) a r e minimax problems of the Chebyshev type, which can be converted

into Bolza problems th rough suitable transformations. Hence, ( P l ) - ( P 8 ) can

be solved employing the family of sequential gradient-restoration algorithms

(SGRA), developed for optimal control problems of the Bolza type.

4.3 . MIELE, A., WANG, T., and MELVIN, W. W., Optimal F l i g h t Trajectories i n

the Presence of Windshear, Part 3, Algorithms, Rice University, Aero-

Astronautics Report No. 193, 1985.

Abstract. T h i s report i s the t h i r d of a ser ies dealing w i t h the

determination of optimal f l i g h t t ra jector ies i n the presence of windshear.

consider the following general problem o f the Bolza type [Problem ( P ) ] : Minimize

a functional w i t h respect t o the s t a t e vector x ( t ) , the control vector u ( t ) ,

and the parameter vector IT which sa t i s fy a system of d i f fe ren t ia l constraints,

i n i t i a l constraints, and f inal constraints.

We

We present the algorithms useful f o r solving Problem ( P ) on a d ig i ta l

computer, more specif ical ly , sequential gradient-restoration algorithms (SGRA) . Both the primal formulation and the dual formulation a r e presented. Depending

on whether the primal formulation i s used or the dual formulation is used, one

obtains a primal sequential gradient-restoration algorithm (PSGRA) o r a dual

sequential gradient-restoration algorithm ( DSGRA) . t The system of Lagrange multipliers associated w i t h the gradient phase

of SGRA and the restoration phase of SGRA i s examined.

i s shown tha t the Lagrange multipliers are endowed w i t h a duali ty property:

they minimize a special functional, quadratic i n the mu1 t i p l i e r s , subject to

the mu1 tip1 i e r different ia l equations a n d boundary conditions, f o r given s t a t e ,

control, and parameter.

For each phase, i t

These duality properties have considerable computational

15 MR-203

impl icat ions: they a l low one t o reduce the a u x i l i a r y optimal cont ro l problems

associated w i t h the gradient phase and the res to ra t i on phase o f SGRA t o

mathematical programming problems involv ing a f i n i t e number o f parameters as

unknowns.

4.4. MIELE, A., WANG, T., and MELVIN, W. W., Optimal F l i g h t T ra jec to r ies i n

the Presence o f Windshear, P a r t 4, Numerical Results, Take-Off, Rice

Universi ty, Aero-Astronautics Report No. 194, 1985.

Abstract. This repo r t i s the four th o f a ser ies deal ing w i t h

the determination o f optimal f l i g h t t r a j e c t o r i e s i n the presence o f windshear.

We obta in numerical r e s u l t s f o r the take-of f problem, employing the dual

sequential gradient - restorat ion algor i thm (DSGRA) , We inves t i ga te a l a r g e

number o f combinations o f performance indexes, boundary condit ions, windshear

models, and windshear i n tens i t i es ,

the angle o f a t tack o r on both the angle o f a t tack and i t s t ime der ivat ive.

The f o l lowing conclusions are reached: (i ) optimal t r a j e c t o r i e s a re

considerably super ior t o constant angle o f a t t a c k t ra jec to r ies ; (ii) optimal

t r a j e c t o r i e s achieve minimum v e l o c i t y a t about the time when the windshear

ends; ( i i i ) optimal t r a j e c t o r i e s can be found which t rans fe r an a i r c r a f t from a

quasi-steady condi t ion t o a quasi-steady condi t ion through a windshear; ( i v )

among the optimal t r a j e c t o r i e s investigated, those minimaximizing Illy1 are

of p a r t i c u l a r i n te res t , because the a1 t i t u d e d i s t r i b u t i o n e x h i b i t s a monotonic

behavior; t h i s i s t rue f o r a moderate windshear and a r e l a t i v e l y severe windshear;

(v) an extremely severe windshear cannot be survived, even employing an optimized

f l i g h t strategy; and ( v i ) the sequential gradient - restorat ion a lgor i thm has

proven t o be a powerful a lgor i thm f o r solving the problem of the optimal f l i g h t

t r a j e c t o r i e s i n a windshear.

I nequa l i t y const ra in ts a r e imposed on on ly

16 AAR-203

4.5. MIELE, A., Summary Report on NASA Grant No. NAG-1-516, Optimal Flight

Trajectories i n the Presence of Windshear, 1985-86, Rice University,

Aero-Astronautics Report No. 195, 1985.

Abstract. This report summarizes the research performed a t Rice University

dur ing the period 1984-85 under NASA Grant No. NAG-1-516 on optimal f l i g h t

t ra jec tor ies i n the presence of windshear.

of motion, problem formulation (take-off), algorithms, and numerical resu l t s

(take-off).

4.6. MIELE, A., WANG, T., and MELVIN, W. W., Optimal Take-Off Trajectories i n

The topics covered include: equations

the Presence o f Windshear, Paper No. AIAA-85-1843-CP, AIAA Atmospheric

Flight Mechanics Conference, Snowmass, Colorado, 1985.

Abstract. T h i s paper i s concerned w i t h optimal f l i g h t t ra jec tor ies i n

the presence of windshear. With particular reference t o take-off, eight fundamental

optimization problems [Problems (Pl )-( PS)] a re formulated under the assumptions

that the power setting i s he ld a t the maximum value and tha t the airplane is

control led through the angle of attack.

Problems (P1 ) - ( P3) a re 1 east-square probl ems of the Bo1 za type. Problems

(P4)-(P8) a re minimax problems o f the Chebyshev type, which can be converted

into Bolza problems through sui table transformations.

solved employing the dual sequential gradient-restoration algori thrn (DSGRA) fo r

optimal control problems.

Numerical resu l t s a r e obtained for a large number of combinations of

These problems a r e

performance indexes, boundary conditions, windshear models, and windshear

intensi t ies . However, for the sake of brevity, the presentation of this paper

i s rest r ic ted to Problem (P6), minimax ] A h [ , and Problem (P7), minimax lAyl.

Inequality constraints are imposed on t h e angle of attack and the time derivative

o f the angle o f a t t a c k .

17 AAR-203

The following conclusions a re reached: ( i ) optimal t ra jec tor ies a re

considerably superior t o constant angle of at tack t ra jec tor ies ; ( i i ) optimal

t ra jec tor ies achieve min imum velocity a t about the time when the windshear ends;

( i i i ) optimal t ra jector ies can be found which t ransfer an a i r c r a f t from a

quasi-steady condition t o a quasi-steady condition through a windshear; ( i v )

among the optimal t ra jec tor ies investigated, those minimaximizing IAyl a re of

par t icular in te res t , because the a1 t i tude dis t r ibut ion exhibits a monotonic

behavior; this i s true for a moderate windshear and a re la t ively severe windshear;

( v ) an extremely severe windshear cannot be survived, even employing an optimized

f l i g h t strategy; and ( v i ) the sequential gradient-restoration algorithm (SGRA)

has proven t o be a powerful algorithm for solving the problem of the optimal

f l i g h t t ra jec tor ies i n a windshear.

4.7. MIELE, A., WANG, T., and MELVIN, W. W., Optimal Take-Off Trajectories

i n the Presence of Windshear, Journal of Optimization Theory and

Applications, Vol. 49, flo. 1 , pp. 1-45, 7986.

Abstract. T h i s paper is concerned w i t h optimal f l i g h t t ra jec tor ies i n the

presence of windshear.

optimization problems [Problems ( P l ) - (P8) ] a r e formulated under the assumptions

tha t the power set t ing is held a t the maximum value and tha t the airplane i s

controlled th rough the angle of attack.

U i t h par t icular reference to take-off, eight fundamental

Problems (Pl)-(P3) are least-square problems o f the Bolza type. Problems

( P 4 ) - ( P 8 ) a r e minimax problem of the Chebyshev type, which can be converted

into Bo1 za probl ems through sui tab1 e transformations.

solved employing the dual sequential gradient-restoration algorithm (DSGRA) fo r

optimal control problems.

Numerical resul ts are obtained for a large number of combinations of

I

These probl ems a re

performance indexes, boundary conditions, windshear models, and windshear

18

~~

AAR-203

intensi t ies .

i s res t r ic ted t o Problem ( P G ) , minimax l A h l , and Problem ( P 7 ) , minimax lAyl.

Inequality constraints a re imposed on the angle of attack and the time derivative

of the angle o f attack.

Howevcr, for the sake o f brevity, the presentation of this paper

The following conclusions are reached: ( i ) optimal t ra jector ies a re

considerably superior t o constant-angl e-of-attack t ra jector ies ; ( i i ) optimal

t ra jector ies achieve m i n i m u m velocity a t about the time when the windshear ends;

( i i i ) optimal t ra jector ies can be found which t ransfer an a i r c r a f t from a

quasi-steady condition t o a quasi-steady condition through a windshear; ( i v )

as the boundary conditions are relaxed, a higher f inal a l t i t ude can be achieved,

a lbe i t a t the expense of a considerable velocity loss; ( v ) among the optimal

t ra jector ies investigated, those solving Problem (P7) a r e t o be preferred,

because the a l t i tude distribution exhibits a monotonic behavior; i n addition,

for boundary conditions BC2 and BC3, the peak angle of a t tack i s below the

maximum permissible value; ( v i ) moderate windshears and relat ively severe

windshears a re survivable employing an optimized f l i g h t strategy; however,

extremely severe windshears are not survivable, even employing an optimized

f l i g h t strategy; and ( v i i ) the sequential gradient-restoration algorithm (SGRA) , employed i n i t s dual form (DSGRA) , has proven t o be a powerful algorithm fo r

so lv ing

4.8.

optimum

the problem of the optimal f l i g h t t ra jec tor ies i n a windshear.

MIELE, A., WANG, T., and MELVIN, W. W., Guidance Strategies f o r Near-

Opt imum Performance i n a Windshear, Part 1 , Take-Off, Basic Strategies,

Rice University, Aero-Astronautics Report No. 201, 1986.

Abstract. T h i s paper i s concerned w i t h guidance s t ra tegies for near-

performance i n a windshear. The take-off problem i s considered w i t h

reference t o f l i gh t i n a vertical plane.

19 AAR-203

F i r s t , t ra jec tor ies fo r optimum performance i n a windshear a re determined

for d i f fe ren t windshear models and different windshear in tens i t ies . Use is

made of the methods of optimal control theory i n conjunction w i t h the dual

sequential gradient-restoration algorithm (DSGRA) f o r optimal control problems.

In this approach, global information on the wind flow f i e ld i s needed.

Then, guidance s t ra tegies for near-optimum performance i n a windshear

a re developed, s ta r t ing from the optimal t ra jec tor ies .

guidance schemes a re presented: ( A ) gamma guidance, based on the re la t ive

path inclination; (B) theta guidance, based on the pitch a t t i t ude angle; and

( C ) acceleration guidance, based on the relat ive acceleration.

local information on the wind flow f i e ld i s needed.

Specifically, three

In t h i s approach,

Numerical experiments show tha t guidance schemes ( A ) , (E), ( C ) produce

t ra jec tor ies which a re quite close t o the optimum t ra jec tor ies .

the near-optimum t ra jec tor ies a re considerably superior t o the t ra jec tor ies

arising from al ternat ive guidance schemes.

In addition,

An important character is t ic o f guidance schemes ( A ) , ( B ) , ( C ) i s the i r

simplicity.

instrumentation and/or modification o f avai 1 ab1 e instrumentation.

4.9.

Indeed, these guidance schemes a re implementable using available

MIELE, A., WANG, T., and MELVIN, W. W., Guidance Strategies fo r Near-

Optimum Performance i n a Windshear, Par t 2, Take-Off, Comparison Strategies,

Rice University, Aero-Astronautics Report No. 202, 1986.

Abstract. In the previous paper, t ra jec tor ies fo r optimum performance

i n a windshear were determined f o r different windshear models and d i f fe ren t

windshear intensi t ies . Then, guidance s t ra tegies fo r near-optimum performance

i n a windshear were developed, s ta r t ing from the optimal t ra jec tor ies .

three guidance schemes were presented: ( A ) gamma guidance, based on the re la t ive

Specifically,

20 AAR-203

path inclination; ( B ) theta guidance, based on the pitch a t t i tude angle; and

( C ) acceleration guidance, based on the reldtive acceleration.

I n this report, several comparison s t ra teg ies a re investigated for the

sake of completeness, more specifically: (D) constant alpha guidance; ( E ) constant

velocity guidance; (F) constant theta guidance; ( G ) constant re la t ive path

inclination guidance; (H) constant absolute path inclination guidance; and

( I ) l inear a1 t i t ude distribution guidance.

Numerical experiments show tha t guidance schemes (D) through ( I ) a re

infer ior t o guidance schemes ( A ) through ( C ) for a variety of technical reasons.

In particular, for the case where the wind velocity difference is AWx = 120 f t sec-’

and the windshear intensi ty is AWx/Ax = 0.030 sec-l, i t is shown tha t the

t ra jec tor ies associated w i t h guidance schemes ( D ) through ( I ) h i t the ground,

while the t ra jec tor ies associated w i t h guidance schemes (A) through ( C ) c lear

the ground.

4.10. MIELE, A., WANG, T., and MELVIN, W. W., Guidance Strategies fo r Near-

Optimum Take-Off Performance i n a Windshear, Paper Ho. AIAA-86-0181,

AIAA 24th Aerospace Sciences Meeting, Reno, Nevada, 1986.

Abstract. T h i s paper i s concerned w i t h guidance s t ra teg ies fo r near-

The take-off problem is considered w i t h opt imum performance i n a windshear.

reference to f l i g h t i n a vertical plane.

First, t ra jector ies for optimum performance i n a windshear are

determined for different windshear models and difference windshear intensi t ies .

Use i s made of the methods of optimal control theory i n conjunction w i t h the

dual sequential gradient-restoration algorithm (DSGRA) f o r optimal control

problems. I n th is approach, global information on the w i n d flow f i e l d is needed.

21

Then, guidance s t ra tegies for near-optimum performance i n a windshear

a r e developed, s ta r t ing from the optimal t ra jector ies . Specifically, three

guidance schemes are presented: ( A ) gama guidance, based on the re la t ive

path inclination; ( B ) theta guidance, based on the pitch a t t i t ude angle; and

( C ) acceleration guidance, based on the re la t ive acceleration.

local information on the wind flow field i s needed.

In this approach,

Numerical experiments show that guidance schemes ( A ) , (B), ( C ) produce

t ra jec tor ies which a re quite close to the optimum trajector ies . In addition,

the near-optimum t ra jec tor ies a r e considerably superior to the t ra jec tor ies

a r i s ing from a1 ternative guidance schemes.

An important character is t ic of guidance schemes ( A ) , (B), ( C ) is their

simp1 ic i ty .

instrumentation and/or modification of available instrumentation.

4.11. MIELE, A., WANG, T., and MELVIN, W. W., Guidance Strategies fo r Near-

Indeed, these guidance schemes a r e implementable us ing available

Opt imum Take-Off Performance i n a Windshear, Journal of Optimization

Theory and Applications, Vol. 50, No. 1, pp. 1-47, 1986.

Abstract. T h i s paper is concerned w i t h guidance s t ra teg ies fo r near-

The take-off problem is considered w i t h optimum performance i n a windshear.

reference to f l i g h t i n a vertical plane.

First, t ra jec tor ies for optimum Performance i n a windshear a r e

determined for different windshear models and different windshear in tens i t ies .

Use i s made of the methods of optimal control theory i n conjunction w i t h the

dual sequential gradient-restoration algorithm (DSGRA) fo r optimal control

problems. I n this approach, global information on the wind flow f ie ld is needed.

Then, guidance s t ra tegies for near-optimum performance i n a windshear

a re developed, s ta r t ing from the optimal t ra jec tor ies . Specifically, three

guidance schemes are presented: ( A ) gamma guidance, based on the re la t ive

path inclination; (B) theta guidance, based on the pitch a t t i tude angle; and

( C ) acceleration guidance, based on the relat ive acceleration.

approach, local information on the wind flow f i e ld i s needed.

In t h i s

Next, several a l ternat ive schemes a re investigated for the sake of

completeness, more specifically: ( D ) constant alpha guidance; ( E ) constant

velocity guidance; ( F ) constant theta guidance; ( G ) constant re la t ive path

inclination guidance; ( H ) constant absolute path inclination guidance; and

( I ) l inear a1 t i tude d i s t r i b u t i o n guidance.

Numerical experiments show tha t guidance schemes ( A ) - ( C ) produce

t ra jec tor ies which a re qui te close t o the optimum t ra jec tor ies . In addition,

I the near-optimum t ra jec tor ies associated w i t h guidance schemes ( A ) - ( C ) are

considerably superior to the t ra jec tor ies ar is ing from the a1 ternative guidance

schemes (D)-(I).

An important character’stic o f guidance schemes ( A ) - ( C ) is the i r

simplicity.

i ns trumenta ti on and/or modification o f avai 1 a bl e i ns trumenta t ion .

4.12.

Indeed, these guidance schemes are implementable us ing available

MIELE, A., WANG, T., and MELVIN, W. W., Optimization and Acceleration

Guidance of Flight Trajectories in a Windshear, Paper No. AIAA-86-2036-CP,

AIAA Guidance, Navigation, and Control Conference, Williamsburg, Virginia,

1986.

I

I

Abstract. This paper i s concerned with guidance s t ra tegies for near-

The take-off problem i s considered w i t h i

op t imum performance i n a windshear.

reference t o f l i g h t in a vertical plane. I n addition t o the horizontal shear,

the presence of a downdraft i s assumed.

F i rs t , t ra jec tor ies f o r optimuni performance i n a windshear a re determined

f o r different windshear models and different windshear in tens i t ies . Use i s

23 AAR-203

made of the methods of optimal control theory i n conjunction w i t h the dual

sequential gradient-restoration algorithm (DSGRA) f o r optimal control problems.

In this approach, global information on the wind flow f i e ld i s needed.

T h e n , guidance s t ra tegies for near-optimum performance i n a windshear

a re developed, s tar t ing from the optimal t ra jec tor ies . Specifically, an

acceleration guidance scheme, based on the re la t ive acceleration, i s presented

i n both analytical form and feedback control form. In this approach, local

information on the wind flow field is needed.

Numerical experiments show tha t the acceleration guidance scheme produces

t ra jec tor ies which a r e quite close to the optimum trajector ies . In addition,

the near-optimum t ra jec tor ies a r e considerably superior t o the t ra jec tor ies

ar is ing from al ternat ive guidance schemes.

An important character is t ic o f the acceleration guidance scheme i s i t s

Indeed, this guidance scheme is implementable us ing available simplicity.

i ns trumenta ti on and/or modi f i ca t i on o f avai 1 ab1 e i ns trumen ta ti on.

4.13. MIELE, A., WANG, T., and MELVIN, W. W.9 Optimization and Gamma/Theta

Guidance of Flight Trajectories i n a Windshear, Paper No. ICAS-86-564,

15th Congress of the International Council of the Aeronautical Sciences,

London, England, 1986.

Abstract. T h i s paper i s concerned w i t h guidance s t ra teg ies f o r near-

The take-off problem i s considered w i t h optimum performance i n a windshear.

reference t o f l i g h t i n a vertical plane. In addition to the horizontal shear,

the presence o f a downdraft i s assumed.

First, t ra jec tor ies fo r optimum performance i n a windshear a re determined

f o r different windshear models and different windshear intensities. Use i s

made o f the methods o f optimal control theory i n conjunction w i t h the dual

~

24 AAR-203

sequential gradient-restoration a1 gori t h m (DSGRA) fo r optimal control problems.

I n this approach, global information on the wind flow f i e ld i s needed.

Then, guidance s t ra tegies fo r near-optimum performance i n a windshear

a re developed, s tar t ing from the optimal t ra jector ies .

guidance schemes are presented: the absolute gamma guidance scheme, based

on the absolute path inclination; the relative gama guidance scheme, based on

Specifically, three

the re la t ive path inclination; and the theta guidance scheme, based on the

pitch a t t i t u d e angle. In this approach, local information on the wind flow

field i s needed.

Numerical experiments show tha t the gamma/theta guidance schemes produce

t ra jec tor ies which a re quite close to the optimum trajector ies .

the near-optimum t ra jec tor ies a re considerably superior t o the t ra jec tor ies

In addition,

ar is ing from a1 ternative guidance schemes.

An important character is t ic o f the gamma/theta guidance schemes is their

simplicity.

i nstrumentation and/or modification o f avail able instrumentation.

Indeed, these guidance schemes a re implementable us ing available

4.14. MIELE, A., CJANG, T., and MELVIN, W. W., Quasi-Steady F l i g h t t o Quasi-

Steady F l i g h t Transition i n a Glindshear: Trajectory Optimization, 6th

IFAC Workshop on Control Applications of Nonlinear Programing and

Optimization, London, England, 1986.

Abstract. T h i s paper i s concerned w i t h the optimal t ransi t ion of an

1 a i r c r a f t from quasi-steady f l i g h t to quasi-steady f l i g h t i n a windstiear. The take-off

problem is considered w i t h reference t o f l i gh t i n a vertical plane. In addition

to the horizontal shear, the presence of a downdraft is considered. I t is

assumed tha t the power set t ing i s held a t the maximum value and tha t the

a i r c r a f t i s controlled through the angle of attack. Inequal i ty constraints

a r e imposed on b o t h the angle of attack and i t s time derivative.

25 AAR-203 I

The optimal t ransi t ion problem i s formulated as a minimax problem or

Chebyshev problem of optimal control : the performance index be ing minimized

is the peak value of the modulus of the difference between the absolute path

inclination and a reference value, assumed constant.

t h e Chebyshev problem i s converted into a Bolza problem.

problem is solved employing the dual sequential gradient-restoration algorithm

(DSGRA) fo r optimal control problems.

By sui table transformation,

Then, the Bolza

Numerical experiments a re performed f o r different windshear intensities

and d i f fe ren t windshear models.

(WS1) i t includes the horizontal shear and neglects the downdraft; (WS2) i t

neglects the horizontal shear and includes the downdraft; (WS3) i t includes

both the horizontal shear and the downdraft.

Three basic windshear models a r e considered:

The numerical resu l t s lead to the following conclusions: ( i ) not only the

t ransi t ion from quasi-steady f l i g h t t o quasi-steady f l i g h t i n a windshear is

possible, b u t i t can be performed i n an optimal way; ( i i ) f o r weak-to-moderate

shear/downdraft combinations, the optimal t rans i t ion is characterized by a

monotonic climb; i n this monotonic climb, the absolute path inclination i s

nearly constant through the shear region; this constant value decreases a s

the shear/downdraft intensi ty increases; ( i i i ) fo r severe shear/downdraft

combinations, the optimal transit ion i s characterized by an i n i t i a l climb,

followed by nearly horizontal f l i g h t i n the shear region, followed by

renewed climbing i n the after-shear region; ( i v ) the re la t ive velocity decreases

i n the shear region, achieves a m i n i m u m value a t about the end of the shear,

and then increases i n the after-shear region; ( v ) i n the after-shear region,

the absolute p a t h inclination increases a t a rapid r a t e a f t e r the velocity

recovery i s almost completed; and ( v i ) the dual sequential gradient-restoration

algorithm has proved to be a powerful algorithm for s o l v i n g the problem of

the optimal transit ion i n a windshear.

26 AAR-203

5. B i bl iography

Flight Mechanics, App l i ed Aerodynamics, S t ab i l i t y and Control

5.1. ABZUG, M. J., Airspeed S tab i l i t y under Windshear Conditions, Journal

of Aircraft , Vol. 14, 1977.

5.2. FROST, W . , and CROSBY, B. , Investigations of Simulated Aircraf t

Flight through Thunderstorm Outflows, NASA, Contractor Report

No. 3052, 1978.

5.3. FROST, W., and RAVIKUMAR, R., Investigation of Aircraft Landing

i n Variable Wind Fields, NASA, Contractor Report No. 3073, 1978.

5.4. FROST, W., Flight i n Low-Level Windshear, NASA, Contractor Report

No, 3678, 1982.

5.5. GERA, J., T h e Influence of Vertical Wind Gradients on the

Longitudinal Motion of Airplanes, NASA, Technical Note No. 0-6430,

1971 ,

5.6. GEM, J., Longitudinal S t ab i l i t y and Control i n Windshear w i t h

Energy Height Rate Feedback, NASA, Technical Memorandum No. 81828,

1 980.

5.7. HAINES, P. A , , and LUERS, J. K., Aerodynamic Penalties of Heavy

Rain on a Landing Aircraf t , NASA, Contractor Report No. 156885, 1982.

HOUBOLT, J., Survey on Effect o f Surface Winds on Aircraf t Design

and Operations and Recommendations f o r Needed Wind Research, NASA,

Contractor Report No. 2360, 1973.

5.8.

5.9. MCCARTHY, J . , BLICK, E. F., and BENSCH, R. R. , J e t Transport

Performance i n Thunderstorm Windshear Conditions, NASA, Contractor

Report No. 3207, 1979.

27 AAR-203

5.10.

5.11.

5.12.

5.13.

5.14.

5.15.

5.16.

5.17.

5.18.

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