Control Applications of Nonlinear Programming and Optimization 1989

124 Pages·1990·13.881 MB·English

by  H.B. Siguerdidjane, P. Bernhard

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CONTROL APPLICATIONS OF
NONLINEAR PROGRAMMING
AND OPTIMIZATION 1989
Proceedings of the 8 th I F AC Workshop
Paris, France, 7—9 June 1989
Edited by
H. B. SIGUERDIDJANE
LSS, CNRS/ESE, Plateau du Moulon, 91192 Gif-sur-Yvette Cedex, France
and
P. BERNHARD
IN RI Λ, route des Lucioles, Sophia Antipolis, 06560 Valbonne, France
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First edition 1990
Library of Congress Cataloging-in-Publication Data
IFAC Workshop (8th: 1989: Paris, France)
Control applications of nonlinear programming and optimization
1989: proceedings of the 8th IFAC Workshop, Paris, France,
7-9 June, 1989/edited by H. B. Siguerdidjane, P. Bernhard.—1st ed.
p. cm.—(IFAC workshop series: 1990, no. 2)
1. Automatic control—Congresses. 2 Nonlinear programming—
Congress. 3. Mathematical optimization—Congresses.
I. Siguerdidjane, H. B. (Houria Β.) II. Bernhard, Pierre.
III. International Federation of Automatic Control. IV. Title.
V. Series.
TJ213.139 1989 629.8—do20 90-32047
British Library Cataloguing in Publication Data
Control applications of nonlinear programming and
optimization 1989.
1. Automatic cintrol. Applications of nonlinear
programming
I. Siguerdidjane, H. B. II. Bernhard, P.
III. International Federation of Automatic Control IV. Series
629.8312
ISBN 0-08-037869-2
These proceedings were reproduced by means of the photo-offset process using the manuscripts supplied by the
authors oj the different papers. The manuscripts have been typed using different typewriters and typefaces. The
lay-out, figures and tables of some papers did not agree completely with the standard requirements: consequently
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changed: nor could the English be checked completely. Therefore, the readers are asked to excuse any deficiencies
of this publication which may be due to the above mentioned reasons.
The Editors
Printed in Great Britain by BPCC Wheatons Ltd, Exeter
IFAC WORKSHOP ON CONTROL APPLICATIONS OF
NONLINEAR PROGRAMMING AND OPTIMIZATION
Sponsored by
International Federation of Automatic Control (IFAC)
Mathematics of Control Committee
Theory Committee
Applications Committee
Association Française pour la Cybernétique Economique et Technique
(AFCET)
Centre National de la Recherche Scientifique (CNRS)
Organized by
Institute National de Recherche en Informatique et en Automatique (INRIA)
Laboratoire des Signaux et Systèmes (LSS) CNRS/ESE
International Programme Committee
P. Bernhard, France (Chairman) M. Fliess, France
H. B. Siguerdidjane, France (Co-chairman) L. Le Letty, France
M. D. Ardema, USA R. W. H. Sargent, UK
J. F. Bonnans, France I. Troch, Austria
R. Bulirsch, FRG V. Utkin, USSR
E. M. Cliff, USA Κ. Well, FRG
National Organizing Committee
J. F. Bonnans
Th. Bricheteau
H. B. Siguerdidjane
PREFACE
This volume contains a selection of papers presented at the workshop on Control
Applications of Nonlinear Programming and Optimization held in Paris on June 7-9,1989.
The aim of this workshop was to exchange ideas between scientists who apply nonlinear
prograrnming and optimization to realworld control problems and those who develop new
methods, algorithms and software.
The attendance to the workshop was forty-nine participants from eight countries, two invited
lectures and twenty two contributed papers were presented. The invited speakers were
Professor A. Miele (USA) and Professor I. Troch (Austria).
The workshop covered various optimization problems of control systems and of numerical
solutions, algorithms and software packages. The applications dealed with the windshear
problems, optimization of aircrafts and spacecrafts trajectories, optimal control for robots,
the optimization of urban traffic control, of control of general mechanical systems , of serial
multi level inventory systems and of a short term hydro thermo-scheduling. Some papers
dealed with robust control.
The workshop benefited from the sponsorship of IFAC (International Federation of
Automatic Control) through the Committees on Applications, Theory and Mathematics of
Control, of AFCET (Association Française pour la Cybernétique Economique et Technique)
and of CNRS (Centre National de la Recherche Scientifique).
All contributed papers included in this volume have been reviewed by the International
Programme Committee with the help of experts, whose valuable contribution we thankfully
acknowledge.
Dr. Houria B. Siguerdidjane
Pr. Pierre Bernhard
vu
Copyright © IFAC Control Applications of
Nonlinear Programming and Optimization,
Paris, France 1989
ADVANCES IN WINDSHEAR RECOVERY
RESEARCH1
A. Miele2
Aero-Astronautics Group, Rice University, Houston, Texas, USA
Abstract. This paper summarizes the research on the optimization and
guidance of flight trajectories in the presence of windshear, performed
by the Aero-Astronautics Group of Rice University during the period 1934-
89. This research refers to windshear recovery systems and covers two
major areas of investigation: optimal trajectories for take-off, abort
landing, and penetration landing; and guidance schemes for take-off,
abort landing, and penetration landing.
Key Words. Flight mechanics, windshear problems, take-off, abort landing,
penetration landing, optimal trajectories, optimal control, feedback
control, guidance strategies, piloting strategies.
1. INTRODUCTION shear encounter; here, the intent is to
Low-altitude windshear is a threat to fly smartly across a microburst, if an
the safety of aircraft in take-off and inadvertent encounter takes place.
landing. Over the past 20 years, some 30 Obviously, windshear avoidance systems and
aircraft accidents have been attributed to windshear recovery systems are not
windshear. The most notorious ones are the mutually exclusive, but complementary to
crash of Eastern Airlines Flight 066 at one other.
JFK International Airport (1975),the crash Examples of windshear avoidance
of PANAM Flight 759 at New Orleans Inter- systems are: ground-based mechanical
national Airport (1982), and the crash of systems (anemometers), ground-based radar
Delta Airlines Flight 191 at Dallas-Fort systems (Doppler radar), and airborne
Worth International Airport (198 5). systems (radar or lidar). Examples of
To offset the windshear threat, there windshear recovery systems are: maximum
are two basic systems : windshear avoidance angle of attack guidance, constant pitch
systems and windshear recovery systems. A guidance, acceleration guidance, and gamma
windshear avoidance system is designed to guidance. At this time, some of the above
alert the pilot to the fact that a wind- avoidance systems and recovery systems
shear encounter might take place; here,the appear to be promising. Further research
intent is the avoidance of a microburst. A is both necessary and desirable in order
windshear recovery system is designed to to l'et the dust settle and prior to making
guide the pilot in the course of a wind- large commitments of funds to one system or
1 2
This research was supported by NASA Foyt Family Professor of Aerospace
Langley Research Center, by Boeing Sciences and Mathematical Sciences, Aero-
Commercial Airplane Company, by Astronautics Group, Rice University,
Air Line Pilots Association, and Houston, Texas.
by Texas Advanced Technology Program.
1
2 Α. Miele
another.For previous research,see Refs.1-48. two basic phenomena are always present:
1.1. Rice University Research on shear and downdraft. Therefore, it is
Windshear. This paper refers to windshear important that these essential charact-
recovery systems and summarizes the eristics be present in the wind model
research performed at Rice University employed in optimization and guidance
during the period 1984-89 under the studies. In this paper, the assumed wind
sponsorship of NASA Langley Research model has the following properties: (a)
Center, Boeing Commercial Airplane there is a transition from a uniform head-
Company, and Air Line Pilots Association. wind to a uniform tailwind, with nearly
This research was initiated in 1983 at the constant shear in the core of the down-
suggestion of Captain W.W. Melvin of Delta burst; (b) the downdraft achieves maximum
Airlines and ALPA. Its objective was: to negative value at the center of the down-
study three problem areas, namely, take- burst; (c) the downdraft vanishes on the
off, abort landing, and penetration land- ground, h = 0; and (d) the wind velocity
ing; for each problem area, to determine components nearly satisfy the continuity
optimal trajectories, namely, trajectories equation and the irrotationality condition
minimizing a suitable performance index; in the core of the downburst.
for each problem area, to develop guidance In this model, the horizontal shear
schemes approximating the optimal traject- (hence, the horizontal wind component) is
ories in real time.
independent of the altitude; and the down-
While the above problem areas are draft (the vertical wind component)
quite different from one another, they are increases linearly with the altitude.
related by common mathematical grounds and Therefore, the wind model has the form
common procedures. The optimization
W = λΑ(χ), W = λ(η/η*)Β(χ). (1)
x h
problems can be solved by means of a
single algorithm, the sequential gradient- Here, the parameter λ = A w/A w^ charact-
x x
restoration algorithm, developed by the erizes the intensity of the windshear/down-
Aero-Astronautics Group of Rice University draft combination; the function A(x) re-
over the years 1970-85. From a compre- presents the distribution of the horizontal
hensive study of the optimal trajectories, wind versus the horizontal distance; the
the dominant properties of these traject- function B(x) represents the distribution
ories can be found. Then,these dominant of the vertical wind versus the horizontal
properties are employed to develop guidance distance; and h is a reference altitude,
A
laws that it is desirable to approach in h* = 1000 ft. Also, AW is the horizontal
actual flight. Finally, these guidance laws wind velocity difference (maximum tailwind
are implemented via feedback control minus maximum headwind) and AW ^ = 100 fps
X
schemes in such a way that the guidance is a reference value for AW .
χ
trajectories approximate the optimal Decreasing values of λ (hence, de-
trajectories. See Refs. 1-17. creasing values of ΔW ) correspond to mild-
χ
1.2. Wind Model. Although no two er windshears; conversely, increasing
windshear encounters are exactly alike. values of λ (hence, increasing values of
Advances in Windshear Recovery Research
AW ) correspond to more severe windshears. assumption that maximum power setting is
Therefore, by changing the value of λ, one employed and that inequality constraints
can generate shear/downdraft combinations are imposed on the angle of attack and the
ranging from extremely mild to extremely angle of attack rate. Hence, the power
severe. setting $ is such that
Figure 1 shows the functions W^(x)
3 = 1, 6= 0, (3a)
and W (x,h), computed for λ = 1, corre-
h
sponding to AW = 100 fps. For other values and the angle of attack α is such that
of λ (hence, for other values of ΔΜ ), the
χ
α <_ α , -ά* £ α £ ά*' (3b)
Λ
ordinates in Fig. 1 must be scaled pro-
portionally to λ (hence, proportionally to where α is the stick-shaker angle of
Λ
attack and α = 3.0 deg/sec is the limiting
Λ
2. TAKE-OFF PROBLEM (REFS- 1"8) angle of attack rate.
2.1. Optimal Trajectories. Optimal With the above understanding, optimal
take-off trajectories were studied with trajectories were computed for three
the aid of the sequential gradient- Boeing aircraft (B-727, B-737, and B-747)
restoration algorithm for optimal control and several windshear intensities. From the
problems. Several performance indexes were extensive computations, certain general
considered; the most reliable one was conclusions became apparent:
found to be the deviation of the absolute (i) the optimal trajectories achieve
path inclination from the nominal value. minimum velocity at the end of the shear;
Because the deviation has a maximum value (ii) the optimal trajectories require
along the flight path, attention was an initial decrease in the angle of attack,
focused on minimizing the peak deviation. followed by a gradual increase; the maximum
The resulting optimization problem is a permissible angle of attack (stick-shaker
minimax problem or Chebyshev problem of angle of attack) is achieved near the end
optimal control, in which the desired of the shear;
minimum performance index has the form (iii) for weak-to-moderate windshears,
the optimal trajectories are characterized
I* = min I, (2a)
α by a continuous climb; the average value of
I = max|Ye - ye |0. 0 < t < τ, (2b) the path inclination decreases as the
intensity of the shear increases;
Y = arctan[(Vsiny + W )/(Vcosy+ W )].(2c)
e h x
(iv) for relatively severe windshears,
Here, t is the running time; τ is the final the optimal trajectories are characterized
time; α is the angle of attack; y Q is the by an initial climb, followed by nearly-
absolute path inclination; y^^ is the value horizontal flight, followed by renewed
at Ye at t = 0 ; γ is the relative path climbing after the aircraft has passed
inclination; V is the relative velocity; through the shear region;
and Wx, Wh are the wind components [see (v) weak-to-moderate windshears and
Eqs. (1)]. relatively severe windshears are survivable
This problem was studied under the employing an optimized flight strategy;
Α. Miele
however, extremely severe windshears are optimal trajectories are merely benchmark
not survivable, even employing an optimized trajectories that it is desirable to
flight strategy; approach in actual flight.
(vi) in relatively severe windshears, Since global information is not
optimal trajectories have a better survival available, what one can do is to employ
capability than maximum angle of attack local information on the wind field, in
trajectories and constant pitch traject- particular, local information on the wind
ories . acceleration and the downdraft. Therefore,
For a particular case, the Boeing the guidance problem must be addressed in
B-727 aircraft with a take-off weight W = these terms: Assuming that local infor-
180000 lb, a flap deflection 6 = 15 deg. mation is available on the wind accele-
p
and an initial altitude h 50 ft, optimal ration, the downdraft, as well as the state
r
trajectories were computed for of the aircraft, we wish to guide an air-
craft automatically or semiautomatically
λ = 0.8, 1.0, 1.1, (4a)
in such a way that the key properties of
corresponding to the optimal trajectories are preserved.
Based on the idea of preserving the
AW = 80, 100, 110 fps. (4b)
properties of the optimal trajectories,
Figure 2 shows the resulting altitude pro- three guidance schemes were developed at
file h(t), velocity profile V(t), and angle Rice University: (a) acceleration guidance,
of attack profile a(t). Clearly, as AW based on the relative acceleration; (b)
x
increases, the altitude profile of the absolute gamma guidance, based on the
optimal trajectory changes: it is entirely absolute path inclination; and (c) theta
ascending for AW 3 0 fps; it includes a guidance, based on the pitch attitude angle.
nearly-horizontal branch for AW 100 fps; Among these, the acceleration guidance and
and it includes a descending branch for the gamma guidance are of particular
AW = 110 fps. Should the horizontal wind interest.
χ ^
velocity difference be further increased Key to all of the guidance schemes is
to AW = 120 fps, the B-727 would crash, the recognition of the fact that, in the
even flying an optimal take-off trajectory. shear region, the dynamical effects due to
2.2. Guidance Trajectories. The shear and downdraft can be combined into a
computation of the optimal trajectories single factor, called the shear/downdraft
requires global information on the wind factor (Ref. 7),
field; that is, it requires the knowledge
F = w/g - W /V. (5)
x h
of the wind components at every point of
the region of space in which the aircraft The larger the value of F, the larger are
is flying. In practice, global information the dynamical effects caused by the shear/
is not available; even if it were avail- downdraft combination.
able, there would not be enough computing In the acceleration guidance (AG), the
capability onboard and enough time to pro- guidance law has the form
cess it adequately. As a consequence, the
Advances in Windshear Recovery Research 5
(AG) V/g + C F = 0; (6) angle of attack guidance (MAAG) and
constant pitch guidance (CPG). See Fig. 3,
therefore, it enforces the proportionality
which refers to the Boeing B-727 aircraft,
between the shear/downdraft factor and the
λ = 1.0, and AW = 100 fps. While the MAAG
instantaneous acceleration. In the gamma
trajectory crashes, the CPG trajectory
guidance (GG), the guidance law has the
scrapes the ground; on the other hand, the
form
AG trajectory is close to the OT. An
(GG) y - γ (1 - C F) = 0; (7) analogous remark holds for the GG traject-
e θ 0 2
ory, which is not shown, since it is nearly
therefore, it enforces the achievement of
identical with the AG trajectory.
decreasing values of the absolute path
2.3. Simplified Guidance Trajectories.
inclination with increasing values of the
As stated above, the previous guidance
shear/downdraft factor. Note that (6) and
schemes require local information on the
(7) apply to the shear region.
wind acceleration, the downdraft, and the
In the aftershear region, guidance
state of the aircraft. While this infor-
laws different from (6) and (7) are needed.
mation will be available in future aircraft,
Specifically, in the acceleration guidance
it might not be available on current air-
(AG), partial velocity recovery is enforced,
craft .
(AG) V - C3VQ =0. (8) For current aircraft, one way to
survive a windshear encounter is the quick
In the gamma guidance (GG), partial path
horizontal flight transition technique,
inclination recovery is enforced,
based on the properties of the optimal
( G) G Y VeO = °- ) (9 trajectories. The quick horizontal flight
e -
transition technique requires an initial
These guidance laws are then imple-
decrease of the angle of attack, so as to
mented via feedback control forms, such
decrease the path inclination to nearly
that the difference α - α is made pro-
horizontal. Then, nearly-horizontal flight
portional, via suitable gain coefficients,
is maintained during the windshear
to the violation of any of the guidance
encounter. Climbing flight is resumed after
laws (6)-(9). Here, α is the instantaneous
the shear is past.
angle of attack and α = α(V) is the
For relatively severe windshears,the
nominal angle of attack.
quick horizontal flight transition technique
Guidance trajectories were computed
yields trajectories which are competitive
for the Boeing B-727 aircraft using the
with those of the guidance schemes dis-
above guidance schemes. It was found that
cussed previously. In addition, for re-
both the acceleration guidance (AG) and
latively severe windshears, the quick
the gamma guidance (GG) produce traject-
horizontal flight transition technique
ories which are close to the optimal
yields trajectories which have better
trajectories (OT). In addition, the result-
survival capability than those resulting
ing near-optimal trajectories are superior
from maximum angle of attack guidance and
to the trajectories arising from maximum
constant pitch guidance.