Wind shear is a dangerous condition where there is a sharp change in the direction and magnitude of the wind velocity over a short distance or time. This condition is especially dangerous to aircraft during landing and takeoff and can cause a sudden loss of lift and thereby height at a critical time. A numerical simulation showed the effective performance of the Linear Quadratic Regulator and the Nonlinear Inverse Dynamics controllers. The major conclusions are listed and discussed

Joshi, Amit S.

English

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FLIGHT PENETRATION OF
WIND SHEAR"
CONTROL STRATEGIES
AMIT S. JOSHI
PRINCETON UNIVERSITY
A Typical Microburst EncountQr
Wind shear is a dangerous condition where there is a sharp change in the
direction and magnitude of the wind velocity over a short distance or time.
This condition is especially dangerous to aircraft during landing and
take off and can cause a sudden loss of lift and thereby height at a
critical time.
I
RUNWAY
107
https://ntrs.nasa.gov/search.jsp?R=19880014343 2020-03-20T06:17:22+00:00Z
Problem Formulation
Wind shearis a conditionofchangingspeedand/ordirectionof thewind
rapidlyover a shortdistance.A microburstis a specialkindof a windshear
in whicha downwardblastof air hitsthe ground.Windshears,especially
microbursts,are very hazardousto aircraftmanueuveringcloseto the
ground. If unopposedbythe pilot,thereis a suddengain in heightandthen
an equallysuddenlossin heightwhichcan leadto a crash.Microbursts
havecauseda numberof crashesduringtake off and landing.
Wind Shear
A change in wind velocity in a brief time so as to cause
a rapid change in the speed of the air flowing over the
wing
Microburst
A downward blast of air which spreads on hitting the
ground
Effects of Wind Shear
Loss in height and or position due to changes in lift
causing severe hazard to the airplane
Problems
Could cause crashes during landing, take off or other
,maneuvers close to the ground
108
Linear Quadratic Regulator
An aircraft represents a nonlinear system; hence the general problem of its
control is nonlinear. It is possible to linearize about the flight Path to
allow control strategie_sdeveloped for linear systems to be applied..1.One
approach is to use a Linear Quadratic Regulator, in which a control law that
minimizes a quadratic cost function is found. Minimization leads to a
linear state feedback strategy giving a stable closed-loop system.
x ==> states = { velocity flight-path-angle pitch-rate angle-of-attack
height thrust }
u ==> controls = { elevator throttle }
w ==> disturbances = { horizontal-wind vertical-wind }
A ( ) ==> perturbation about nominal
Q, M, R ==> cost weighting matrices
This approach hinges on choosing good cost weights. This is clear
from the results. The choice can be difficult, and it is dependent on
the aircraft; this is a major drawback of this method.
Given linear system
& _ = F&x+ G&u+ L&w
Define a " cost"
px
0
Minimizing J leads to
Au = -R1 (GTs+ _ ),_ = -CAx
Where S is the solution of the Riccati Equation
{F-GR'_S+S/F-G.'I_ r - SGR"Es+o-..'MT=o
109
Variation of Cost Weights
The cost weights for the cost function were obtained by using a
combination of state-rate and direct state and control weighting. No
direct cross weighting between the states and the controls was used.
The cost weights were varied by varying the direct cost weights on the
controls.
A sinusoidal model of the microburst developed by Mark Psiaki was
used.It had a maximum headwind/tailwind of 10.7 m/s and a maximum
downdraft of 6 m/s.
u
State rate weighting used" 100 O10
10
1
O 100
1
B
Q0 (direct statecost weighting) = Identity
R0 (direct control weighting ) =
['::o] ,:ol
Microbursts :
Range • Headwind/tailwind begins 0 m and ends 3000 m
Strength • 10.7 m/s
Range • Downdraft begins 1050 m and ends 1950 m
Strength • 6 m/s
110
Gains with Varvina Cost Weiahts
By varying the costweight associated with the elevator, differentgains were
obtained for a Linear Quadratic Regulator. The aim of this figure is to show
that different weights lead to very different control requirements.
Gains set #1:
F" -1
/ -3.407E-2 -9.228 -7.403 -4.496 -2.801E-2 -8.644E-2 /C L /4.268E-1 3.260E+1 1.926E+1 1.299E+1 1.468E-2 1.983
Gains set #2:
F -7c, 6336 308E  66E,
l /3.372E-1 8.2 9.741E-1 1.104 7.082E-2 1.762
Gains set #3 :
F 7
/ -1.783E-1 -6.367E+1 -2.585E+1 -2.503E+1 -3.108E-1 -3.309E-1 /C / /4.100E-2 6.429E-1 1.806E-3 8.513E-3 5.802E-3 3.414E-1
111
Effects of Cost Weighting
Variations on Altitude 6nd Controls
Simulation results for a simplified model of a Boe727 (representing a
typical jet aircraft) follow. All the simulations are for a take off
condition. The nominal flight conditions are
Airspeed = 71.628 m/s flight path angle = 0.0523 rad
pitch rate = 0.0 rad/s angle of attack = 0.0611 rad
altitude = 3.0 m thrust = 0.8713" maxthrust
elevator =-0.0518 rad throttle = 0.8713 * max thrust
These conditions apply to allthe simulation results. SI units are used
on all the plots.
The altitude vs range plot shows the dramatic improvement due to the
control law over the openloop performance. It also shows that a
choice of high elevator cost weight leads to a poorer performance.
The difference between the throttle and the thrust plots is due to
modelling of a lag between the generation of the thrust and the
throttle command. The throttle vs range plot shows that as the cost
weight of the throttle is increased, the throttle saturates later. The
result is that the control activity of the elevator goes up. In general,
though, the saturation of the throttle sooner or later means that the
elevator is the main control remaining. This is seen from the similarity
between all the elevator plots.
112
o= open loop
A= gain set #2
Effects of Cost Weighting
o= gain set #1Variations on Altitude and Controls
• = gain set #3
1.0 10
000.0 Altitude vs Rang, e ,.o Thrus_ vs Range
450.0 I.O
400.0 1.6
350,0 1.4
300.0 1.2
. cl
200,0 0.0 _
150.0 0.6
I00.0 0.4
00.0 0.2
I I I I I I_ar_ 9e I I I l I
0.0 0.0
0.0 50.0 I00.0 150.0 200.0 350.0 300.0 350.0 400,0 450.0 500.0 I0.0 0.0 50.0 IOLI.O 150.0 200.0 2500 300.0 350.0 4000 4_0,0 S000 IO0
113
Effects of Cost Weights
Variations on Velocity and Angular Control_
These set of plots show the velocity, angle of attack(alpha), flight
path angle (gamma), and the pitch rate as functions of the range.
Again there is a dramatic improvement over the open-loop case. The
flight path angle and velocity variations are reduced considerably,
leading to the improvement in the altitude vs range seen earlier. The
pitch rate and the angle of attack are affected much less. It can be
seen that the angle of attack reaches quite high values.
a= open loop
A = gain set #2
o = gain set #1
• = gain set #3
|.0 1.0
050 - ,, 00 !
00.0 : _ 0.6
: ,
81.0 _ O.A
'_ $0,0
_ 0,0 _0,0 I000 I_0.0 200.O'_--2_e_T 3000 350.0 4000 4",00 50_sO XO0
2) 70.0 -0.2 o
05.0 -0.4 -
60.0 -0,6
55.0 / -0.8I P I I Rar, _9e I I I I i I Range
50.0 ° -I.0
0.0 60.0 I00.0 150.0 200.0 250,0 300.0 350.0 400.0 450,0 500.0 I00
sam I I
1.0 1,0
,.o Gamma us R_e o, PiCch-race vs R_nge0,6 0,3
0.4 0.2 i
, 0.0 50_.0 200.0 250.0 300J ' I I I I G" 0.0 I _0._0 100.0 150:_.0 ZS0.O 300J) 350.0 400.0 450.0 ,_.00, Iq050.0 |00.0 350.0 400.0 450.0_ 500,0 |0.0
-O.Z _ ! -0.1
-0.4 t
_,. -o.2
-0.6 -0,3
-0.8 I "0.4Ran9e Range
-I.0 -0.5
114
Nonlinear Inverse Dynamics
The aircraft is a nonlinear system, and we can tackle the nonlinear
control problem directly using a nonlinear inverse dynamics
( NID ) approach.
In this approach we asume that we have a nonlinear system with
x, u, w the states, controls,and the disturbancesas for the LQR.
Additionally we define an output _ as linear combination of the states.
Since an exact inverse of the nonlinear system cannot be found, we
calculate an approximate one.
We differentiate the output 'd' times until some controls aDpear in each
of the outputs. It is assumed that there is a function defining the
desired output. Setting the derivatives of the output equal to those of
the desired output .gives us a set of nonlinear algebraic equations.
When solved for the control with some simplifications, we get the set
of controls giving the desired output behavior,
Given a nonlinear system
f(x ,_.u
y..= Cx
Differentiate y.. 'd' times until the control
appears in the output to get
[d]
y_. = g(X,V_)
where v = ,u,u,u...)
Now let the desired output be .y..
desired
Now set "_u= _ .... O and solve
_d]= [d]g (X_,U ) = _lesired
115
Nonlinear Inverse Dynamics
In the NID approach, we are free to choose either a functional form of
the desired output or choose the dth derivatives of the desired output
and then specify a dynamics for the output.
Choosing ,V-desiredwe choose the desired
dynamics for y
[d]
Alternatively choosing a dynamics for ,y..
[d] desired
gives us the ,y..
116
(_onclusions
* The results of the simulation show the effective performance of the
LQR and the NID controllers. The major conclusions that one can
draw from these results are
* The LQR seems to try to keep the variation in the velocity and the
flight path angle to the minimum. This is obvious in hindsight when it
is realized that these are the major factors controlling deviations from
the desired flight trajectory.
* There is more variation in the angle of attack and the pitch rate
from the nominal values.
* Thrust almost always saturates for the LQR type of control law and
a reasonably large microburst. There is an initial reduction in the
thrust as the aircraft enters the microburst ; then there is sharp
increase until it saturates. Finally the thrust comes back to normal as
the aircraft gets out of the region of the microburst.
* The elevator shows a very different behaviour. High cost weights
associated with the elevator lead to lesser elevator use. However, with
the thrust saturated, the elevator is effectively the only control and this
shows in the elevator behaviour.
Linear Quadratic Regulators lead to a good
performance with a good choice of costs.
LQR s can require very high feedback gains
for a good performance.
Nonlinear Inverse Dynamics with complete
solution of nonlinear equations promises
to give excellent performance.
NID with complete solution of nonlinear
equations would have the penalty of:
1) Time required
2) Possibility of none or multiple
solutions.
117


Flight penetration of wind shear: Control strategies

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