exploring the multidimensional steady flight envelope of a business jet aircraft silvia ferrari...
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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