Neural Network Prediction of New Aircraft Design Coefficients

NASA

Technical

Memorandum

112197

Neural Network Prediction of New Aircraft Design Coefficients Magnus N_rgaard, Institute of Automation, Technical University of Denmark Charles C. Jorgensen, Ames Research Center, Moffett Field, California James C. Ross, Ames Research Center, Moffett Field, California

May 1997

National Aeronautics

and

Space Administration Ames Research Center Moffett Field, California 94035-1000

NEURAL

NETWORK

PREDICTION

OF

NEW

AIRCRAFT

DESIGN

COEFFICIENTS

Magnus

NCrgaard,*

Charles

C. Jorgensen,

and James

C. Ross

Computational Sciences Division Ames Research Center

SUMMARY This paper tunnel

discusses

tests.

predictions validation,

Using

a neural a hybrid

network neural

tool for more effective

network

optimization

aircraft

method,

design

we have

evaluations produced

during

wind

fast and reliable

of aerodynamical coefficients, found optimal flap settings, and flap schedules. For the tool was tested on a 55% scale model of the USAF/NASA Subsonic High Alpha

Research

Concept

(SHARC)

lift, drag, moment

of inertia,

settings.

The latter network

optimal

flap schedules.

aircraft.

Four different

networks

were trained

to predict

and lift drag ratio (C L, C D, C M and L/D) from was then used to determine

an overall

optimal

coefficients

angle of attack

flap setting

of

and flap

and for finding

INTRODUCTION

Wind tunnel testing can be slow and costly due to high personnel utilization. Thus, a method that reduces the time spent in a wind associated with a test, are of major Modem wind tunnels have become

interest to airframe highly sophisticated

of performance

designs.

determination lift/drag

features

of aircraft

of the coefficient

ratio (L/D)

but the techniques

Currently,

a new design

researchers inspection

to interpolate of curves.

describe

the complex

approach

potentially

computing

methods

new approach earlier

calculations.

test is followed between

relationships

between

The longer

manual

evaluation

the procedure variables.

automation

term benefits

to consider

testing

of the aircraft

design

approach.

Spin-off

of measurement

are a significant

neural

benefits

processing reduction

as well. To allow

is based

on visual

expressions

networks

aerodynamic

(C u) and

we emphasize

and analysis.

is to f'md mathematical

Although

only

moment

In this paper tunnel

data fitting

this task, e.g., numerical

a very cost effective increased

chosen

and flap settings.

by extensive

and design engineers. used to measure a number

of drag (CD), pitching

to other steps of wind

measurements,

able to perform provide

In this study we have

of angle of attack are applicable

One way to automate

and include

manufacturers test facilities

of lift (C,_), coefficient

as functions

prediction,

overhead and intensive power tunnel, as well as the work load

to

are not the only

simulations,

such soft

can also result

from

and aids for checking

in costs and faster

*This author was at the NASA Ames Neuro-Engineering Laboratory in 1994 as part of a cooperative student work study program between NASA and the Institute of Automation, Electronics Institute, and Institute of Mathematical Modelling, at the Technical University of Denmark. The Danish Research Council is gratefully acknowledged for providing financial support during his stay.

a

development

of new aircraft,

automotive

or alternate

tunnel uses such as more aerodynamically

efficient

design.

This paper

is organized

and one powerful Next, we describe networks

A short introduction

to Multilayer

Perceptrons

(MLP)

is given

method we used (a variation on the Levenberg-Marquardt method) is presented. how a subset of test measurements were used with the technique to train four

to predict

settings.

as follows:

aerodynamical

We then present

coefficients

two applications.

and the L/D ratio, given

The first addresses

angle

the problem

of attack

and flap

of determining

an

"overall optimal" flap setting using a method based on integration of L/D vs. C L. The second demonstrates an easy strategy to find optimal flap schedules. Finally, details of the software tool set are given

in an appendix

as a supplement

to documentation

MULTILAYER

in the project

code.

PERCEPTRONS

The phrase "neural network" is an umbrella covering a broad variety of different techniques. The most commercially used network type is probably the MLP network. See reference 1. An example a MLP network is shown in figure 1. In this study we used a two-layer network with tangent hyperbolic activation functions in hidden layer units, and a linear transfer function in the output units. A two-layer network is not always an optimal choice of architecture (goodness measured terms of the smallest number of weights required to obtain a given precision), but it is sufficient approximate any continuous faster in this case.

function

A MLP network

type of an "all-purpose"

shown

is a special

an excellent

corresponds

ability

for function

to the following

1. A three

represent

input,

the biases.

yi(w,W)=

Here

two output, f/(x)

function,

in Sj/Sberg,

to implement

recent shown

and

situations in figure

has 1

w_

two layer MLP

Wio

network.

(ref. 4). One disadvantage

The weights

from

the inputs

set to 1

and Fj (x) = x.

_-- Fii

Wijf

j

/n

__aWjlZll-Wjo l=l

feature offered by this type of network well, without requiring an extravagant et.al.,

in many

(ref. 3). The network

k.j=l

A special functions

which

is easier

in to

form

= tanh(x)

%hj(w)+

fi

well (ref. 2), and training

approximation

functional z_

Figure

arbitrary

of

is that it can be trained amount of parameters

compared

to other network

I / "[-Wio

(1)

to approximate many (weights). This is discussed types

is that training

is slow

becausethe networkimplementsa non-linearregression,i.e.,thereis a nonlinearrelationbetween theadjustableparameters, the weights,andthe output. In this study,we wereinterestedin enhancinggenericneuralnetworksfor wind tunneltest estimation.Obtainingnetworktrainingdatais very costly,e.g.,$3,000dollarspertunnelhourfor theNationalFull-ScaleAerodynamicComplexatAmesResearchCenter.Consequently,only limited datasetswereavailableandusuallyasbyproductsof previouslyscheduledtests.The sizeof the data setimposesanupperlimit onhow manyweightsthenetworksshouldcontain.In practice,since thereis alsouncertaintyassociated with themeasurements, thenumberof datapointsmustexceed thenumberof weightsby a sufficientlylargefactorto ensurethatgoodgeneralizationmay be achieved.Trainingtime,onthe otherhand,is not of primeimportance.Many argumentscanbe madein favor of someform of MLP networksasthefight choicefor the givenproblem.The real problemis obtainingextremelyhigh accuracies criticalfor commercialviability. TRAINING Thetrainingphaseis the processof determiningnetwork measurement

data. The treatment

of different

we will consider

a generic

quantity

of three

variables:

Angle

different

aerodynamical

'y' instead. of attack

weights from a collected set of coefficients is essentially identical,

If the aircraft

(c¢), leading

flaps

edge

are coupled,

flap angle

so

y becomes

a function

(LE), and trailing

edge flap

angle (TE). y = g0 (¢p)

(2)

where

LE rE The function

'g' is unknown,

but the wind

tunnel

(3) tests provide

us with a set of corresponding

y - _0

pairs i

i

."

ZN = {[q9 ,y ],t = 1..... N} Naturally the measurements number of different sources.

(4)

of y are not exact, but will be influenced in undesired ways from a All measurement errors are grouped in one additive noise term, e y = g0(q_)+ e

The objective

is now to train the neural

network

(5)

to predict

y from

= _(tp) The predictor Expressed functions

is found

precisely, contained

from the set of measurements,

we wish to determine in the chosen network

a mapping architecture

Z N --> b

(6) Z N, from here on denoted from the set of measurements g(_0; 0)

the training

set.

to the set of

(7) 3

sothat _ is close

to the "true"

y. 0 is the parameter

this case, the network

weights).

A common

for goodness

definition

V(O)=

Thus,

training

becomes

1 --_ 2N

a flexible

training

in the neural

algorithm

containing

all adjustable

of fit in neural nets is the mean square ::

parameters

i=1

(in

error

1 N = _...._.. ___., (g(0))2

_i(0))2

(yi_

(8)

/=1

a conventional

back-propagation,

vector

unconstrained

but somewhat network

optimization

ad hoc gradient

community.

search

Ease

problem.

For various reasons

method,

has been the preferred

of implementation,

utilization

of the

inherent parallel structure, and the ability to work on large data sets are the main arguments justifying the use of this method. However, in the present case where the data sets are of limited size, backpropagation

is not the best choice.

Marquardt many

method

for solving

ways superior

Instead

we have decided

the optimization,

to back-propagation method,

optimization

packages

(ref. 5). Some important

convergence

to a (local)

necessary method

except

important

properties

our objective networks

of its neural

numerical a network

advantages

architecture.

in making

was to create

a user-friendly,

a generic

The Levenberg-Marquardt

method

has numerous

mappings variations.

guaranteed inputs

out in Mor6

Such

tool, which

for application

the complex

as pointed

is crucial

(ref. 6) the are

in this case since if in fact neural

in nonlinear

The simplest

are

advantages

use and determine

required

The

horse of many

are speed,

user-specified

these benefits.

easy-to-apply

methodology

of performing

Moreover,

to achieve

it is in

methods.

is a work

of the method

and minimal

Levenbergapproaches

search

implementation,

robustness,

free of ad hoc solutions

were capable

the original in reference

independent

for providing

is surprisingly

gradient

as well as most other gradient

Levenberg-Marquardt

minima,

to use the so-called

since like conjugate

aero design.

strategy

may be found

in

contribution of Marquardt, while one adaptation to neural network training is discussed 7. The version used here belongs to the class of trust region methods found in Fletcher

(ref. 8). Just as back propagation,

the Levenberg-Marquardt

algorithm

is an iterative

search

scheme

(9)

_k+_) = O_k)+ ld

5) V(O(k))--V(O(k)+hq'))