Exploring the Multidimensional Steady Flight Envelope of a Business Jet Aircraft

Silvia Ferrari

Princeton University

FAA/NASA Joint University Program on Air Transportation,

FAA Tech Center, Atlantic City, NJ

October 26-27, 2000

Table of Contents

• Introduction and motivation

• Proportional-integral neural network controller

• Nonlinear business jet aircraft model

• Trim map definition

• Exploring aircraft multidimensional flight envelope

• Computation and storage of trim data

• Trim control surfaces

• Conclusions and future work

Introduction

• Stringent operational requirements introduce: Complexity Nonlinearity Uncertainty

• Gain scheduling: Design a set of “local” linear controllers Interpolate local operating point designs

• Gain scheduling limitations: Nonlinearities dominate Fast transitions between operating points

• Nonlinear control design takes advantage ofknowledge gained from linear controllers

Neural Networks for Control: Copying with Complexity

• Applicability to nonlinear systems

• Applicability to multi-variable systems

• Parallel distributed processing and hardware implementation

• Learning

• Flexible logic

Motivation for Neural Network-Based Controller

• Network of networks motivated by linear control structure

• Multiphase learning:

Pre-training

On-line training during piloted simulations or testing

• Improved global control

• Pre-training phase alone provides:

A global neural network controller

An excellent initialization point for on-line learning

Proportional-Integral Controller

ycx(t)

ys(t)

CI

CF

CB

Hu Hx

+

+

+

+

-

-

-

u(t)PLANT

ty~

Closed-loop stability: ,ct xx 0~ ty ,ct uu

yc = desired output, (xc,uc) = set point. ,~

cs tt yyy

ycx(t)

˜ y t

+

+

+

+

-

u(t)uc

+

-

uB(t)

uI(t)

xc

ys(t)

e

a

NNI

NNF

NNBSVG

CSG

PLANT

tx~

tts uxh ,

Where: ,ct xx ,0~ ty ,ct uu cs t yy

Proportional-Integral Neural Network Controller

Nonlinear Business Jet Aircraft Model

Aircraft Equations of Motion:

For the model:

tttt

ttttdt

td

upxhy

upxfxx

,,

,,

State vector:

Control vector:

Parameters:

Output vector:

r

m

n

t

t

t

t

y

p

u

x

Teee rqpzyxwvu x

TSTRA u

XB

ZB Thrust

Drag

Lift

V

mgp

q

r

YB

Body axis: XB, YB, ZB

Aircraft Body Frame and Flight Path Angles

Flight path angles: , ,

XE

.

YE

.

V

not shown on this figureCommand input: Tccccc V y

Aircraft Angles

Euler angles: , , Airflow angles: ,

YB

XB ZB

V

ZB

YB

XB

u

w

u

V

V

BE

rqp

LKinematic equations:

: non-orthogonal transformation matrixBEL

Neural Network Modelling of Aircraft Trim Maps

uc

NNF

yc

0,,,,:, 00 ccTcccT andPXU upxfpxupx

are solved for the control settings, uc, over

initial estimate of operational domain, (X0, P0).

• The trim equations,

,0,, puxf ccT

• The aircraft flight envelope consists of

all (x,p), for which a solution uc, s.t.:

• Command State Generator (CSG):

employs kinematic equations to produce xc

ucyc

p(a,yc

)

TUxc

CSG

a

a = [ h ]

Example of Coordinated Maneuver

YB

XB

ZB

mg

acent

V

acent = centripetal acceleration

= rate of turn

Steady Climbing Turn:

[Etkin, “Dynamics of Flight”]Command input: Tccccc V y

0 50 100 150 200 250 300 350 4000

2000

4000

6000

8000

10000

12000

14000

16000

Estimated Business Jet Aircraft Steady Flight Envelope

Stall speed:

CL = CLmax

Based on simplified aircraft model, and aerodynamics and thrust characteristics

Thrust-limited minimum speed

Maximum allowable dynamic pressure

Compressible Mach number limit

[Stengel, “Flight Dynamics and Control”]

h (

m)

V (m/s)

Spherical Mapping of Aircraft Angles

• Spherical trigonometric relations:

,,,,,,,,,,,,,,,,,

fnfnfn

• Redundant set of angles: ,

,

cos

coscossincossincossin

,cos

sinsincossincossin

assuming = .

• Angles defined on a unit sphere:

[Kalviste, 1987]

Non-Symmetrical Trim Solution at One Operating Condition

In a coordinated maneuver: .,0,0 const

• Aircraft’s kinematic equations:

coscossincos

sin

rqp

Solve the trim equations,

, for 0,, hrqpwvuh c

TcT yyf

c

c

c

ccc

STRA

uand,,

,,,, fn

Given the command input, , and the scheduling variable, h, Tccccc V y

Subject to,

• Spherical trigonometric relations:

• Airflow angles definitions:

cossinsin

coscos

VV

V

wvu

50100

150200

250

-10

0

10

0

5000

10000

15000

ComputedThree-Dimensional

Flight Envelope

= = 0

V (m/s) (deg)

h(

m)

h(

m)

(deg)

V (m/s)

ComputedMulti-Dimensional

Flight Envelope

= 5, = -20

h(

m)

(deg) V (m/s)

= =0

h(

m)

(deg) V (m/s)

Similarly for all:

maxmin

maxmin::

Rotations of Three-Dimensional Envelope View

h(m)

(deg) V (m/s)

V (m/s)

(deg)

V (m/s)

h(m)

h(m) h(m)

(deg)

V (m/s)

= 5, = -20

(deg)

Multidimensional Flight Envelope Storage Using Cell Arrays

Cell arrays are containers that can hold other MATLAB arrays

000,15

000,2000,10

150908580

1559590

150145

V-H envelope (nested cell)Multidimensional envelope

Optimal Parameters Storage Using Cell Arrays

To each operating point, there corresponds a vector of optimal

parametersOptimal parameter cellMultidimensional cell

V

h

23.001.002.0

31.001.003.0

4.001.003.0

43.002.004.0

27.002.003.0

45.002.003.0

8.003.004.0

60 80 100 120 140 160 180 200 220 240 2600

2000

4000

6000

8000

10000

12000

14000

16000

Computation of Trim Optimal Parameters Throughout Flight Envelope

Vmin(h) Vmax(h)V (m/s)

h (

m)

Algorithm

Search for Vmin

Search for Vmax

Form a vector,

v(h) = Vmin : V : Vmax

Store v(h), *au

Compute:

ua*[, , , v(h)]

h

= 0, = -50

120140

160180

200220 5000

60007000

80009000

100003.8

4

4.2

4.4

4.6

4.8

120140

160180

200220 5000

60007000

80009000

100008.41

8.42

8.43

8.44

8.45

8.46

8.47

100150

200250 5000 6000 7000 8000 9000 10000

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

120 140 160 180 200 220 40006000

8000 10000-3

-2.5

-2

-1.5

-1

-0.5

0

Example: Business Jet Aircraft Trim Surfaces

V (m/s)

R(deg)

A(deg)

T(%) S(deg)

= 5, = -10

h(m)

V (m/s) h(m)

V (m/s) h(m) V (m/s)

h(m)

Summary and Conclusions

Non-symmetrical trim study:

• Full, nonlinear, coupled aircraft dynamics

• Trim solution at one operating point

• Multidimensional flight envelope computation

• Multidimensional envelope and trim data storage

• Results: trim control surfaces