is governed by the engine dynamics. One of the objectives is to implement the controller architecture on the "Stone-Soup-. Simulator". (SSS) as used by NASA.
Development Propulsion
Final
of An Intelligent Control System
Report
October
1, 1999
September
1, 1998-
August
31, 1999
Research supported by the NASA NASA Grant No. NAG 2-1174
Dr. A.J.
Calise
Principal
Dr. R.T.
Rysdyk
Investigator Research Assistant
B.K. J.J.
Leonhardt Totah
J. Kaneshige
School Georgia Atlanta,
of Aerospace Institute GA,
Investigator
NASA
Grant
Monitor
NASA
Grant
Monitor
Engineering of Technology
30332-0150
Ames
Research
Center
Flight
Contents Jlt
List
111
of Figures
iy
Summary Introduction
2
3
4
1
1.1
The
NASA
1.2 1.3
Nonlinear Transition
program
............................
Adaptive Propulsion Control of Results ...............................
Input Output 2.1 Introduction 2.2
Approximate
2.3 2.4
Online Linear
Linearization with .................................... Feedback
Neural Neural
.....................
Neural
Linearization
Network Network
1
Network
and
Bounded
3 4 Augmentation Tracking
Augmentation ..................... Structure ........................
Controller Design 3.1 Architecture .................................... Construction
of the
throttle
positions
3.3
Construction
of the
Pseudo
Signals
...................... .......................
Stability Analyses 4.1 Introduction ....................................
4.3
Roll
Augmentation
4.2.1
Linear
4.2.2
Analysis
4.2.3
Effect
analysis
...........................
of zero dynamics
of output of engine
Effect
tracking
dynamics
of engine
Augmentation
of Yaw
4.4.1
Analysis
4.4.2
Closed
4.4.3
Effect
of Engine
Analytical
Results
5.1
Linear
representation
5.2
Roll
5.3
Pitch
5.4
Yaw channel
rate
dynamics Channel
tracking
.....................
dynamics
..................
16 16
..................
........................ ........................
dynamics
17 18 19
..........................
Dynamics
......................
........................
21 22 22 22 23 23 24 25
of the
augmentation
rate
8 9
17
........................
of zero-dynamics loop
5
17 17
Longitudinal channel ............................... 4.3.1 Analysis of output tracking dynamics 4.3.2
4.4
channel
.........
5 5
13 13
3.2
4.2
5
PCA
augmentation augmentation
MD-11
.......................
.............................. ............................. ............................
25 25 26 28
a,P
6
Stone 6.1
6.2
Soup
Simulator
Implementation
29
Dynamic Inversion for Control Law Design ................... 6.1.1 Linearized AircraftModel ........................
29 29
6.1.2
Symmetric
............................
32
6.1.3
Asymmetric
..........................
32
6.1.4
Engine Control Transformation
Control Control
.....................
33
Implementation .................................. 6.2.1 Model Inversion Assumptions ......................
34 34
6.2.2
Control Architecture ...........................
35
6.2.3
Simulation Results
39
7
Recommended
8
Conclusion
Further
............................
54
Developments
57 58
Appendices A
McFarland
B
Data
Inertia
for the Boeing
58
Coefficients
59
B747-400
6O
References
ii
List 1.1
PCA
conventional
locity,
control
altitude,
location
c.g.
of operational
3.1
Architecture
for angular
4.1
Roll
channel
setup
5.1
Roll
channel
root
5.2
Root
5.3
structure,
rate
locus,
are
scheduled
configuration,
nature
using
tracking
Cleft) and MD-11
channel
propulsive error
without model,
6.4 6.5 6.6 6.7
angle channel
path
angle
_f=0deg)
Inner-Loop
Rate
H=2,000ft,
Sf=0deg)
6.11
_f=0deg)
Rate with with
Flight
Path
Azimuthal Flight
Outer-Loop
Azimuthal
modelled helicopter,
error
compensator,
c) Flight
error
compensator,
c) Flight
.............................
40 41 in Pitch
Channel
(V=lS0kts, 43
Attitude and
Hold Response
in Roll Channel
(V=lS0kts, 44
Pitch
Rate
Doublets
(V=lS0kts,
H=2,000ft,
Pitch
Rate
Doublets
(V=180kts,
H=2,000ft,
45
and and Pitch Flight Angle
46 Pitch
Rate
Doublets
(V=285kts,
H=30,000ft, 47
Rate
Regulation
(V=180kts,
H=2,000ft,
$F=O 48
Path
Angle
Regulation
(V=lS0kts,
H=2,000ft, 49
Step
(Heading)
Path
Angle
Response
Angle
Step
(V=180kts, Response
H=2,000ft, (V=lS0kts,
_f=0deg).
51 Step Response
(Heading)
Angle
Step
(V=285kts, Response
H=30,000ft, (V=285kts,
_F=0deg).
is similar
actuator
in the high bandwidth to the effect
52
H=30,000ft, 53
of the effect of unmodeled rotor
50
H=2,000ft,
...................................... control
29 36
38
b) Tracking
......................................
Outer-Loop
Illustration
. .
...................................
6.15
_F=0deg)
dynamics.
......................................
6.14
7.1
of engine
......................................
to 20deg)
Outer-Loop
Rate
Roll
Deployment
Outer-Loop
(right)
.....................................
6F=O 6.13
with
................................ Rate
Roll
Flap
6.12
b) Tracking
filter,
Command
Deployment
to 20deg)
and
26
......................................
Simultaneous Flap
dynamics.
................................
Roll
Simultaneous
6f=0deg) 6.10
(left)
20
.............................
autopilot
H=2,000ft,
Simultaneous
13
27
roll channel autopilot loop ....................... Rate Command Attitude Hold Response
_f=20deg) 6.9
autopilot
filter,
a) Command hold
Simplified Inner-Loop
_F=0deg) 6.8
a) Command
hold
engine
without
Pitch Roll
.......
..........
................................
6.2
path
of failure, 1I
control
(right)
Yaw channel, simple root locus analysis of the effect Overall Control Archetecture ...........................
6.3
ve-
..........
dynamics
6.1
channel
with:
2
for longitudinal
of the
longitudinal
dynamics
mode,
commands
with
Gains
..........................
designed
for design
loci for the engine
diagram.
operating
engines
Neural
network
law block
location,
2.1
the
of Figures
of unmodeled
,=!
111
dynamics. control engine
The
effect
of the R-50 dynamics
of the un-
autonomous for the IFPCS.
55
Summary The
initial
(IFPCS) tive
design
flight
using
and
demonstration
is documented. control
sources
gain
• The
IFPCS
over • The
provides
is based
This
to provide
scheduled
initial
a bark
range
IFPCS
is applicable
design
up flight
enhances
control.
System
The
flight
IFPCS
adap-
safety
by
enhances
the
ways; system
that
results
in consistent
responses
failures.
to a variety
the
and Control of a nonlinear
IFPCS
in flight
control
Propulsion
implementation
of the
in significant
of unanticipated
reduces
Flight
on the
redundancy
approach
a wide
• significantly
of an Intelligent
design
architecture.
propulsion
conventional
The
of aircraft
laborious
research
models
without
& design
redesign
necessary
and,
in a gain
scheduled
approach. The
control
augmentation
is detailed
setting. The availability differential thrust.
within
of propulsion
an
only
Earlier Propulsion Control Augmentation for a trajectory controller with pilot command is aimed
at
qualities
under
where
demonstrating a variety
propulsion
hard
the IFPCS
PCA
The
The
adaptive
formance. ters
performance
flexibility broad
the
demonstrate work
of the
range
Future with
design
control
bounded
Results
under
a wide class
of uncertainty.
adaptive
will specifically
Robust
and
control
robust that
dynamics
tracking.
and utilization
capability
under
of actuator
a greater
and
phase
associated
as demonstrated
and
in
of the NASA
robust
provides output
results
per-
parame-
disturbances.
demonstrate
performance
Lyapunov
the under
a
like analysis.
of conventional
variety
robust
and adaptive
and
The
of available
saturation.
in flying
design
designs
errors,
to provide using
and
behaviour.
stability
integration
the effects
However,
and input
is proved
iv
engine
all states,
of the ability
will include
augmentation,
address
the
initial
to the gain scheduled
of uncertainties
stability
of the IFPCS
use of propulsion
means
consistency the
of aircraft
symmetric
by NASA provided heading. This work
documents the
Linearization
inputs,
in providing
at best.
estimates
in the quality
architecture
development
that
qualities
demonstrates
stability
is measured IFPCS
report
similar
Input-Output control
work performed of glidepath and
This
confirm
of results
architecture
(PCA) input
IFPCS
handling
uses crude
robust
two
of the
scenarios.
in poor
is capable
IFPCS
In this work,
remain
Robust
is used.
result
simulation,
flexibility
of failure
only
nonlinearities work.
the
approximate
provides
control
lift and
of failure
drag
scenarios.
surfaces devices, Further
to
1 The
objective
of this
controller
architecture
controller
does
redesign.
It provides
range
work
is the design
that
provides
not require and
flight
and
demonstration
control
augmentation
tuning
and
control
and
extensive
backup
of uncertainties
Introduction
disturances,
of an intelligent using
is applicable
propulsive
consistent
unanticipated
to provide
is limited the
reasonable
by throttle
augmentation One
is governed
of the
Simulator"
objectives
(SSS)
tegration
onto
associated
are
in both
by the engine
is to implement
as used
explored.
rate
by NASA
The
controller This
architecture
provides
the
Flight
Control
Simulator
SSS in its current
version
does not provide
The
Following
the
NASA
a similar
DC-10
Encourage separate
from
there
flight
the
engines
path
for emergency throttle and
(AFCS)
nonlinear
from
flight
of engines to safely
used
angle
commands. Ref.
[2]. The
gains
altitude,
• c.g.
location,
• operating
the
flight
An example
• velocity, •
alone
in the
mode:
due
to the
in Fig.
for in-
at Ames.
model The
The
code
environment
for
representative
results
can
of a
of integration
the
dynamic
be used
control
as applied
response
of the
investigated investigations
to augment
or replace
used
specific
driving
to a Boeing with
the
747-400 respect
power
system
[1].
for effectively Manual
These
computer,
1.1 are scheduled
control
aircraft
for newly
of motive
system.
(DFRC)
DFRC
recommended;
systems
source
conventional
of aircraft.
aircraft.
NTSB
control
an alternate
control
Center
the
flight
onboard flight
in a variety
land
"Stone-Soup-
environment
a convenient
dynamic
in 1989,
for the
method
Research
control
utilize
for the
primary
difficult Flight
Iowa
of backup
that
used
a disabled
Dryden
control system
flightpath
input
source
is no satisfactory with
City,
development
airplanes
that
of
program
at Sioux
and
is extremely
NASA
control
research wide-body
the ation.
accident
certified
In general
PCA
control
bandwidth
on the
proper
civilian transport aircraft was used for the analyses in this report. into the SSS code are included as a seperate chapter.
1.1
the
dynamics. the
Ames.
Therefore,
a wide
of propulsive
Furthermore,
Advanced analyses.
without
over
applications aim to of using propulsive
application
and position.
the
and
The
failures.
the
with
development
handling
saturations
only.
of aircraft
response
This report describes the lateral and longitudinal applications. The provide redundancy to the conventional control surfaces. The possibilities control
inversion
forces
to a variety
maintains
including
model
controlling
throttle aircraft
the
use
control in this
aircraft
control
laws
engines
in response
is presented to
situ-
of propulsion
demonstrated the
of
that primaD'
for bank
angle to pilot
in Fig. 1.1, taken
J
I,
_, _
I
k_:nC
I
I
WlXTI
Figure locity,
1.1:
PCA
altitude,
operational
conventional c.g.
location,
control operating
id law
block
mode,
engines.
2
diagram. configuration,
Gains nature
are
scheduled of failure,
with: location
veof
-
unusual
atitude
-
cruise
-
ILS-approach,
recovery,
flight,
• configuration: -
flaps,
-
landing
• nature
gear,
of failure:
-
jammed
-
floating
vs. control
surfaces
and, • number The
and location
experience
for good
gain
of the
designers,
scheduled
designs
scheduling
with
provided
by the
failure
respect
condition The
to the but
may
are not
The
control
include
always
that
in
on a proof
report
but
the
allowed
However,
the safety
that
operator
knows
gain
may
be
which
failure.
flight control system
benefits
from
It can relieve
approximate using
can be found
provides
The
of gain
based
limits that
desirable responses
Adaptive
theoretical
ments
only
of the particular
loss off lifting
approach.
approach
a minimum A rigorous
not
models,
techniques.
will be
over a wide
greatly range
of
surfaces, avoiding
or asymmetricaly the laborious
the reliance
jammed
research
on accurate
control
necessary
simulation
for
models,
available.
control.
gain scheduled
partial
in this report
architecture
significantly
also anticipation
the design
design
presented
as redundant with
of failures
simulation
control
failures.
Nonlinear work
to accurate linear
It requires
if it can provide
Furthermore,
1.2
as access
applicability of a back-up
failures
for control.
on classic
nature
is current,
a gain scheduling which
as well
structure.
unanticipated
surfaces.
available
based
controller
increased
In flight
of engines
Propulsion
aims
at the implementation
a flight safety
nonlinear
in design
Control
adaptive
of a PCA,
of a nonlinear
enhancement
by using
architecture
by allowing
improves for a large
adaptive
the propulsive upon
variety
the
flight sources
conventional
of in flight failures
scheduling. development linearization Lyapunov in Ref
and
is provided.
The
nonlinear
adaptive
like stability [3].
analysis.
theory The
combines
control. proof
The is not
recent
develop-
development included
is
in this
1.3
Transition
For
syntheses,
nonlinear
we
model
MD-11
is from
Aircraft
Model
of Results
used
of a generic (RCAM) flight
includes
actuator
aircraft
weights
capable
of providing
nonlinear
aircraft
design
techniques
engine
nonlinearities,
220,000-331,000
lbs.
a thrust
and
is configured
It has
to weight
are about
version
within is referred
pertaining
to the
of the
aircraft
allows
for good
However, tially, this
flight
design
practice
linearization
that
(SSS).
the
nature
was
the
controller
Ames
research
wings,
The the
together have
Ames. access
stability
and
offered
the use of the most was
later
obtained
model
inversion
by utilizing
derivatives in this report
inversion
data
model.
available.
control
Boeing-747
Ini-
design
for
stability
&
inversion.
architecture facilities
accurate
used for model
and im-
This portable
& control
a crude
a
to information
presented
using
no
of symmetry.
RCAM
of NASA
reliable
The The
engines
to the limited
of the controller
assumptions,
suggests
linear
Due
to obtain
The
mild
industry.
the plane
characteristics,
miniACFS-code
of the RCAM
approximation [6] within
at NASA
of the
not practical under
0.35.
Civil
to promote
aerodynamics.
its main
26 j't outside
these
Research
designed
aeronautical
underneath
as a of the
various aspects of the PCA application. environment were subsequently adapted
Simulator
in the code.
representation
as well as nonlinear
cg, and
as well
on the European an initiative
of approximately
the With
version Soup
it was
performance,
A better
derivatives
It is expected lated
SSS,
prudent
code.
portable
to as Stone
represented
a Jacobian
control
the
is based
MD-11,
linear
in the European
ratio
landing.
of the The
two engines
6 j't below
for
model
represents
and
convenient basis from which to investigate The results obtained within the Matlab plemented
model
model
control
angle,
aircraft
a linear
in MatLab-environment.
[5]. This aircraft
RCAM
The
involving
airliner
Ref. [4]. The
the use of robust
inclination
simulation
will be further in the
4
second
phase
evaluated
in piloted
of this project.
simu-
2
Input Output Linearization with Neural Network Augmentation
2.1
Introduction
The
controller
earization
architecture
(FBL).
contributions
have
fundamentally and
uses
state
In exact until
to the
FBL, the
provided
r, for a given theory
can
Hauser
[10], and
Approximate
extended FBL
of the
an approximate
error.
We augment
the
the approximate
2.2
Approximate
control.
degree.
This
state
dynamics. Linear
Lin-
Fundamental
[7, 8, 9].
method
is
transformation
FBL
control
values
affme
algebraically
techniques
can
FBL
FBL
exact
vari-
of required
derivatives,
the
appearance
sometimes of the
control
first
variable.
Linearization,
The
FBL
formalized
by
in Ref [11].
of a conventional
the
than
(or the independent
number
Feedback
systems
advantages
to time
The
However,
of Approximate
within
rather
to compensate
with respect
for reasonable
form
dynamics, model,
(NN)
one.
a linear
on the
to non-control
network
of the
into
is its relative
combines
aircraft
Using
including
on Feedback
decades.
as it is an exact
linearization
are differentiated
in the
is based last
system.
or negligable
be used
in the
text-books,
local
dependence
variable
of the research
linearization,
system
transformed
is weak still
than
dynamic
part
of attention
in several
rather
explicit
output
control
a lot
conventional
the outputs
first
for this
attracted
from
nonlinear
of the
model
been
feedback,
the
be applied
able)
proposed has
different
transforms then
FBL
paradigm
algebraic
with an adaptive
linearized
of the
expressions
full
approximate
nonlinear
results
system.
in an inversion
'learning-while-controlling'
neural
for this error.
Feedback
Linearization
and Bounded
Tracking Consider
the aircraft
dynamics
represented
y We
in the general =
.f(x,
=
h(x, u)
form
u)
(2.1) (2.2)
wish to design a tracking controllerfora smooth and bounded
trajectorywith bounded
derivativesof sufficient order. Suppose that in the i_ element of the output the components of _
are allsmall. We
reorganize those elements of the output function as Yi = ¢,,0(x)
so that until
eel,0
< < ¢i,0. With
the control
effect
+ e¢i,0(x,
this representation,
is significant.
Each
the output
output
5
(2.3)
u) may be differentiated
yi is differentiated
ri times.
successively Starting
with
Eqn. (2.3),
We say that
the effect
that
are significant
that
¢iz
=
;
.=.
y(') i
+ _¢_,,,__ (x, u,_,,.-.) = ¢_,,,__(x) - ¢_,_,(x, u) + e¢_,_,(x,u, u,.-.)
•
of the control
are
included
+
is significant
in ¢ and
those
when that
8_ = O(¢).
(2.4)
At each
are not are included
step,
in e¢.
the This
terms means
includes,
• the
time
• small
derivative
terms
Definition
2.1
of
from
¢/_.i-1,
the time
A system
to have
neighbourhood
derivative
of ¢iZ-1.
of the form i y
is said
¢,,_(x)+ EC_,,(x, u, ,:,)
t_
= =
a well-defined
f(x,u) ¢0(x)
x E IR n, u, +E_0(y,u)
approximate
(fl,, x fl_,) of a point
local
vector
y elR _ (2.5) relative
(Xo, Uo) if the Jacobian
degree _
[rx, r2,...
is nonsingular
,rm] T in the Vx e f_
and
Vu • f_t,, where
¢(x, u) = [¢1,_1"-" ¢_,,, ]r and,
where
Theorem
ri is the least order 2.1
For a system there
exists
time
Approximate
of the form
derivative
input-ouput
(2.5)
of yi on which
control
the effect
of control
is significant.
linearizability
that has a well-defined
a state-dependent
(2.6)
transformation
approximate
relative
degree
[rl, r2,...,
_,*= [_f _ ... _, lF = V(x, _) that
transforms
the system
into
m decoupled
rm] T,
to a pseudo-control
approximately
(2.7)
linear
systems
y[,-1) = v;+ _¢_,,,(x, u,_,,.-.) (2.8) y_-) = v_ + _¢_,,.(x, _, u,...) A proof
is presented
Consider imate
relative
a system degree
in [11]. of the
form
is defined
(2.5) that
is approximately
linearizable.
The
total
approx-
as W%
= ]Er, i=l
(2.0)
If f =
n, then
to-state
the
closed
loop can
by transformation
be interpreted
to state-variables
as approximately
linearized
from
input-
_, where
_, = [¢,,0(x)... ¢_,,,_l(x)]T
If f < n, then The
normal
the
form
nonlinear of the
system
system
(2.5)
may
can be transformed
be represented
(2.10)
into
a so-called
normal
form.
as
_, = [_,,2... _,,,, ¢',(_,_,,u)+_¢_(¢,#,u,u,.-.)] T = With
e < n, there
these
internal
consider and
the
so-called
the
(2.13)
dynamics,
may
depend are
u is such
which
which
maintained
that
the
must
nonlinearly
zero-dynamics,
derivatives
control
la)
internal
dynamics
all its time
Thus
are
e(_,
(2.12)
be examined
on the
are
the
normal
internal
at zero
by the
zero--dynamics
are
for stability. outputs,
_.
dynamics control
formally
when
u, which defined
In general,
Therefore, the
output
implies
the
2.1
internal
fields,
It is not necessary dynamics
and using
Linearized
the
expression
Model
For convenience
to obtain
by completing
define
for u(x)
Inversion
where,
similarly
=
in terms
of the
pseudo
[y_r,)...
transformation
for #. Eqn
for _ -
We may
(2.1_),
with
also analyse n -
f vector
O.
y_=)]r
of definition
(2.15) 2.1 and
theorem
2.1 may
be sum-
_r
=
_" +E_(x,u)
(2.16)
_,"
=
¢(x,u)
(2.17)
2.6,
¢(x,u) If (I)(x, u) is invertible
expression
_r,
linearizing
to Eqn
(2.14)
transformation
that provides
0.
Design
{, The approximate marized as
the explicit
the state
y,
_ =
by
= e(o, _,) Remark
we
with
respect
= [¢,,,, ... ¢=,,_]_
to the
control,
we may
(2.1s)
express
the
original
control
input
control
u= ¢-_(x,v) Considering the nonlinear nature may be particularly complex are
that we wish to allow impractical to obtain. 7
(2.19) for the If the
plant, the inverse of (I)(x, u) condition of Theorem 2.1 is
satisfied, then the Jacobian ¢oi, = o¢[ _ zo,_ois nonsingular for a set (Xo,Uo) • (9tzx fl_), and a linearizedform of 2.19 may be constructed as
u = uo+ ¢o_{v- ¢o- Cos(x- ,,o)} where uo and Xo are trim values of the plant, ¢o
(2.20)
-- ¢(xo, Uo), and ¢o_ is the Jacobian
with respect to the state at (Xo,Uo). Using Eqn 2.20 for the control law implementation will resultin an inversionerror.If we definethis inversionerror as
zxC_,uo,,,,,,) = ¢(x,u)-{Co+Co,(x-_o)+Co_(U-_o)} (2.21) Remark trim
2.2
values,
Note the
that Eqn
Jacobians,
(_._0)
may
in fact
and the nominal
be implemented
operatin9
with
approximations
for
point
- = _o+_o.'{_-_o-_o,(x-_o)} The
approximations
should
Specifically,
the sign
effort.
be designed
'reasonably'
of the elements
the
accurate
of the Jacobian
to avoid should
(2.22)
unnecessary
control
be known,
A
(2.23) Using
Eqn
(2.21),
Eqn
(2.17)
can be expressed
as
(2.24)
u = ¢(x, u) - _(x0, uo,x, t,) and
the
closed
loop
as _
-- t_ + AE(x0, u0,x,v)
(2.25)
where =
Ae(xo,Uo, X,V)
2.3
Online
In the first phase the
NNs
outputs,
have of the
We design
the
Network
output.
a single application pseudo
The
NN
outline
control
_,_ is an adaptive
t,, is a robustifying
signal.
Construct
channels.
a single
signal
contribution
neural network provided
for multiple
to include
u where
Augmentation
of this work we use a separate
a single
allowing
extension
Neural
(2.26)
A(Xo, Uo,X, V) + 6gt(x,u)
here
(NN)
for each
is stated
In future
NN for multiple
for NNs
work
channel. with
Thus, multiple
we will consider
channels
the
[3].
as --
v_-
from vpd
v_-
an online =
[vpd,
ur
(2.27)
approximator ...vp_
]r, where
to be detailed if i = 1,-..,
next, m
and
where stable.
the coefficients kip chosen Define Ai E IR (ri-l)×('i-1)
Zi
such that the error and bi E _t {r'-l)×(1)
0
1
0
0
...
0
0
0
1
0
...
0 bi
k_l _(P) = _P) - _(P), and
"..
these
definitions,
the
""
be represented
also define
--
elements
in state
[y
y..-
of Eqn
space d_
2.4
Linear
This
section
error
and
a possibly
nonlinear
form
inversion
0 < Ilell < g. The
error.
W*
functions.
• is referred of constant
Here
n2 gives
the
are not
error
selection
[12]. Since
of the
the
input
is a function
of the
NN through
a squashing
The
dynamics
error
output
of
'ideal'
NN
reconstruction
parameter
unique.
values
error, that,
/9 is an hi-vector
or residual
in some
consisting
sense, of basis
of NN outputs. network
I_ is an nl × n2--estimation The
tracking
(2.33)
to as the
= vector.
of the
NN approximation
= ] - e
necessarily
number
For a linear-in-the-parameters
where
boundedness
error
vector
values
guarantees
a linear-in-the-parameters
is an nx × n2-matrix
I1 11.These
minimize
(2.32)
Structure
W*'/_ where
(2.31)
+ vr - A_i)
law that
Consider
are
as
+ bi (v_
adaptation
weights.
which
ui + A_i
Network
a network
network
=
Aiei
(2.30)
_-_r,-l)IT
(2.25),
canonical =
Neural
derives
of the
(2.29)
--
kr,-1
y(") may
are exponentially
:
k_2
e/
With
y_'_ = vp_
T
.:
Define
dynamics as
of W*,
elements
and
_ is made
of "2 is done
to the
NN
of the
NN, UAD,
function.
v)
includes
This
can be written
the a fixed
insures with
through
that
(2.34) up of selected careful
pseudo point
control problem
at least
elements
assessment signal,
of the inversion u,
occurs,
one fixed
of the state
point
which
u is input solution
in turn to the exists.
n2 = 1 as
= Ae + b(VAD -- ]) 9
(2.35)
with A = and
e = [ f _ _ ]' and
b -
[0 1]'.
Define
-Kp
--KD
the
weight
error
as W = I/V - W °. Rewrite
(2.35)
as
d = Ae + bW'/_ The
update
law of the
NN weights
+ b(W*/3
is designed
- ])
(2.37)
as
w = w = -r_ r = 1_ > 0 and
where
( is a scalar
filtered
error
(2.38)
term,
-- e'Pb
(2.39)
with p=
Remark
2.3
nel, i.e. where
xp + _ 2K--7
The implementation
a design
with
second
l_+_e___ = 2rpro
as presented
order
tracking
/:'11PI2 1'12 P_
provides
error
for A CAH
dynamics.
This
(2.40)
in the longitudinal design
may
chan-
be generalized
e = [_(")_(,-1) ... _ ], and b = [ 0 ... A',,P + PAn Remark may
0 1 ]' a size-n
2.4
Notice
also be generalized
architecture
vector,
- 2In = O, with
that
that
A,
and
the closed
integrates
P an n x n-matrix
in canonical
to n2 > 1. the
form
loop system This
three
(2.41)
would
control
that solves
similar defined
be the channels.
to (2.36) by Eqns
but expanded (2.87},
case if one NN In
that
case
(2.38},
is used we
to order
can,
and
n.
(2.39)
in a controller for
instance,
organize
e = [i'_,']' where
x and i
are size n_-vectors.
A will then
A = with A12 = I,_
where
the
index
(2.42)
be designed
[o A21
(2.43)
A22
and
corresponds
A21
=
-diag{Kpa,K_,...,Kp,,_},
Az2
=
- diag{ K o_ , KD2 , "" , KD.2 }
to the elements _
of ft.
Furthermore,
E = [(1 (2 "'"
and
b= [ 0
,
W = W = -FEE' where
a 2n2 x 2n_ block matriz,
_n2]'.
10
(2.44)
l
Neural The
Network
Input
NN can consist
of any linearly
proximately reconstructing because these functions rameterized. their
However,
design
with
centers,
multi-
Mach
number,
used.
The
term.
and
because
Fig 2.1 shows
weights they
are
a large
associated (binary)
number
vectors.
in the
trans-sonic
constants.
structure
very
basis
region,
these
of the state
that
is capable
values
product are the
at interpolation
were
are
used
variations
needed
of ap-
are
variables,
the
network.
of input
variable
network
variations
difficult
in
to represent
sigma-pi
pseudo The
signal
between for networks
to capture
a single-layer
of a sigma-pi
kronecker
poor
functions
implementation,
depiction The
are
of such
consist
a nested
RBFs
In Ref [14], RBFs
current
network
a general
with
that
input
In the
to the
feedforward
the inversion error. Ref [13] uses Radial Basis Functions (RBFs) are universal approximators even when the network is linearly pa-
functions.
inputs
parameterized
it is well know
dimensional
by polynomial
Design
network
control values
and a bias
represent
categories,
and
is the
therefore
weights.
U'o, ecker Tf" product
1
Figure The
2.1:
input/output
Neural map
network of the
structure,
designed
for longitudinal
NN for the longitudinal
channel
channel.
is represented
as
(2.4s)
=
where v_, and v_ are respectively the pitch channel and roll channel elements of the pseudocontrol v. The vector of basis functions, _ is akin to the regressor vector in adaptive control texts.
The
inversion are
basis error
constructed
functions can
are
made
be accurately
by grouping
up from
a sufficiently
reconstructed
normalized
at the
inputs 11
into
rich
network three
set of functions output.
categories.
The The
basis first
so that
the
functions category
is
usedto model the inversionerror effectsdue to changesin airspeed,since control
derivatives
are strongly
dependent
on dynamic
the stability
pressure
C, : { b,, VT, V_ } Where
b, indicates
control
as well
and
pseudo
a bias
as in the
control,
term.
In allowing
states,
v. These,
the and
the
inversion a bias,
plant
error
(2.46) to be nonlinear
is a function
are therefore
the
third
category
transformation
is used between
to approximate the body
higher
frame
and
and
of both
contained
C2 : {b=, xi, xj,..-, The
in the
uncertain
state
order
and
the
vector
C3 by means
of basis of the
functions
kronecker
is composed
xi,
category (2.47)
effects.
inertial
These
may e.g.
be due
to
frame
c3 : {b3,=_} Finally,
in the
variables,
second
v}
the
and
of combinations
(2.4S) of the elements
of C1, C_,
product = kron
(kron(C,,
C2), C3)
(2.49)
where,
kron(=,y) = [=,y,x,y=.-. =,,,y,,]'
12
(2.5o)
3 This
section
systems
describes
in pitch
command.
Controller the
and
roll,
It is assumed
approximately
Design
linearizing
as well as a yaw rate that
the only available
controller
command control
design
system consists
for rate
as alternative of thrust
from
command to roll rate
the
outboard
engines.
3.1
Architecture
Pc
command filters
qc
Aircraft
qc
Figure
3.1:
Architecture
Figure
3.1 displays
attitude-hold
(RCAH)
on the
symmetric
spectively
the architecture in pitch
and
a moment
for angular
and
asymmetric
about
controlled by the application Let the aircaft state be
the body
rate commands
using
propulsive
for the propulsive-control-only roll.
The
inversion
(or differential) pitch
of asymmetric
and
with
respect
application
yaw axes.
setup
The
to the
of thrust. roll rate
control. for rate-commandthrottles These
is therefore
indirectly
(3.1)
may represent the rigidbody aircraftdynamics as
= Ill(X)A(x)/3(x) f4(x)A(x) A(_)}T With
re-
thrust.
x = [vqr_,v_] T We
is based create
the
following
inertia
(3.2)
terms,
c3 "- I_I_- I_.' c4 -
IzIz- _.' c7 -
13
I_' co -
IzIz- _.
(3.3)
let F,
where
A, and,
L, M,
_ represent
and
the aerodynamic
N represent
--f, /2 /3 /4 h ME
ric thrust. symmetric
The
choose
represent
effect
thrust
moments
about
channel,
"_
=
in the
ME
analyses
0
+
A
c7
0
0
c9
÷
moments
as outputs
the input-output
axes.
0
0
0
0
due
Then
c4
o
NE ] ME
to symmetric
for input-output
is mostly
a secondary
thrust,
we neglect
(3.7)
and
asymmet-
linearization
the second
derivative
_pP÷
0+
expression
has
using
-
we write
transmission
may
To obtain
of the inertial
significant coupling
c4
of yl.
_+F,'b
(3.9)
for Yl as
a significant cspr
effect.
the effect
effect
on the first
+ c2p) ME
(3.10)
derivative
c6 (p2 _ r _) + A + c7 ME the
system
in normal
_,.=[P]q Then,
fixed
F
F,c_ NE + (clc9 + c2c4) q NE + (clr
Z12 For the following = [p p q]T and let
body
+
the
on roll rate
consider
appears
the
as input.
asymmetric
therefore
of thrust
pitch
c4L+c_N
roll-rate
thrust
using
Yl In the
=
2)
and
Y, The
czM
respectively
pitch
of asymmetric
3.7, and
=
as
(3.4) (3.5) (3.6)
F,/m fdm F,/,n NE
in Eqn
A
constructed
+
We
of roll-rate
csL÷c4N
cspr-c6(p_-r
and
control
=
(Cl r ÷ c2 p) q
and asymmetric
effect
r
the usual
.-f6.
where
moments,
14
[7].
Therefore
define
(3.12)
be presented
_r = "_C'(x) ÷ g(x)
form
(3.11)
as
[ ME ]NE
(3.13)
Let ".':" representan approximation, then (3.14) The feedbacklinearizing transformation is
]
(3.16)
NE j and
ME
NE may
be obtained
in terms
of the
pseudo
ME The
inverse
by inversion
of 3.16.
vii
of G is
[
= where
control
1]
_ - d_(O)
(3.18)
(c_r+c2p)c7 - d_(_)
det (¢) det(_)
Matrix
G becomes
singular
= - ((c,q
+ F,)c_ + (c2q + Fp)c,)
for an unlikely
pitch
rate,
cgFr + c4Fp
q =
(3.19)
c7
>_ 0.5rad/sec
(3.20)
ClC 9 -_- C2C 4
In order to analyze pled
linearized
on these
modes,
modes
the
remaining
in the
Q reduces
lateral
dynamics and
longitudinal
and
channels.
we consider
If we choose
the decou-
the design
asymmetric
throttle
S_
c-V 0
0
=
1
rr
symmetric
_ is controlled,
based
to _-1
The
when
c_ +r_
(3.21)
c4
deflections,
resp.
s6_,
and
aS_, are obtained
as
(3.22)
=
where 2A6th
--
1
(3.23)
0
2Fr G6t h +2FpF6th
The
closed
loop
stability
analyses
are
contained
15
in section
4.
3.2 Prom
Construction the signals
aSth and
S_h
of the
throttle
the throttle
positions
positions can be constructed
as
1 (s_th+ a_th)
_thl =
6t_ = 51 (s_th -- a_th)
3.3 The
Construction transformation
of the
3.15-3.17
allows
for Rate-Command-Attitude-Hold
Pseudo
the
design
in the
Signals
of desired
pitch
channel
pseudo
signals.
The
can be constructed
as
vq = (tc + el _ + ao
where
_ = qc - q, and
roll channel,
we require
where
q_, _, are obtained
a 2 "d order
Vl6
"-
using
command
Pc
J¢-
_2P
•
filter
Jr-
_IP
16
_ dt
to generate
C_0
f'
signal
(3.26)
a 1't order
Jr"
pseudo
pdt
command
filter
3.1.
In the
Pc, /_, and/5_
(3.27)
4 4.1
Stability
Analyses
Introduction
Most
air-transport
aircraft
roll divergence.
In the
Given
a constant
result
in a particular
aircraft
trim
'frozen'
or 'jammed' of the
This namics
control
miniACFS,
simply
surfaces there
positions the control
provides attitude
stability
the hold
analyses
Roll
4.2.1
channel
Linear
form
of the
and
remain
frozen
control
lateral
and
zero dynamics
laws
investigate
the
and
from
the
the with of the the
the
of SSS
control
time of the
associated using
representation
for a straightforward
cases surfaces.
techniques
linearized
will provide
F¢
without
P
zero
dy-
to provide propulsion of the means
0
+
engine
dynamics,
be
2 fi6,h
A x + b a_th
control we obtain
may
2 Y6,_ 0
¢
0
the
2 F$, h
Y, tan 0o
may be analyzed
of F6,h and
disturbance,
control
its feasibility
on a Jacobian
dynamics,
(4.2)
input in roll is F6,h/fl6,h for the lateral channels =
effect
will
'failure
linearization
7"
1
the
speed.
positions
of zero dynamics
Yp
The
a certain
between
floating
SSS code
approach
rv
The effect of asymmetric in Eqns 2.11, and 2.12,
display
In working
free
by the
÷
"-
may
Augmentation
analysis
The Jacobian linearized represented as
differ
positions PCA
based
This
an external
surfaces.
input-output
control,
are performed
surface
for simulating
of the
(RCAH)
for
of control
may
floating'
surface
We apply
aircraft in a landing configuration. to analyze the zero dynamics.
4.2
'free
provided
analyses
channels.
but
is trimmed given
characteristics,
and
that
that,
stability,
speed.
is no provision
trim
speed
combination
implies
stability
the
chapter
The
stability
and
an aircraft
each
to this unique
implies
for the different
rate command only.
Speed
return
attitude
direction,
as well as the
we used
surfaces' failure.
with
configuration,
speed.
point,
version Therefore,
longitudinal
aircraft
will eventually
The
are designed
by using
Y_,h we may
approximately
=
r, -tane0_r,
310
75
- .-i--. .--.. -. -i.-.-.*..-...i.-..-..i......i......::......i......i.....i......::.i.. illii ::.i i ...._......_......!......!......i.....i......i......i............i.....i......!......i......!......i.... 1 I 0
I 5
I 10
I 15
I 20
I 25
_ 2o E
I 30
I 35
I
40
I 45
I 50
I 55
I
..... ,...... ....... .............. :..... ,___-
I 60
I
i 65
1 70
75
I
_"........ :...... ...... :...... ....... :...... :....
:
......
O-lO
_s
-5
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
Time (s)
Figure 6.6: Inner-Loop H=2,000ft, 5f=0deg).
Rate Command
Attitude
44
Hold Response
in Roll Channel
(V=180kts,
0.015, :_ 1_ =
,
0011' / 0.0051-
rr
.... ....
__1
,
,
i
,
i ...... i...... i...... i ..... : : : : _ ,L_ :-; -;-':- ;.: ":"" ....
,
,
,
i ..... : _.....
i ..... : : [ : .:..[ _ .:. __
,
,
: ..i ..... : \: .:H _ .\: ......
l
i ..... : i .....
i .... _ : I--:.... ,.
I
I
I
:\
:
:
/"
:
:
:
"_---.. _:
:
: -"
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
!
!
!
!
!
!
!
!
!
!
!
!
!
!
,
Roll Rate Pitch Rate . .
:
:
:
' i I: _ .... :
....; .... ;..... :...... i...... i...... i..... ;.... -5 -60 _
.... i ........... ...... ...... ..... iiiiii:iii:iiiiiiiiiiiiiiiiiiiiiiiiill
-851-""_":
..... :..... "...... :...... :...... : .... "
-_5t- :i..._I::
.....!......i......i......_.....!....-t
-901- _!
..... i......!....-I
-5
0
5
10
15
20
25
30
35
4
,
i
i
i
i
i
i
!
40
-_ 3 .... i ..... _...... i...... _..... i..... i ..... : ...... i............ v
45
50
55
i
i
i
60 1 ......
65
70
i
i
75
! ..... ! ...... ]...... i...... ! ..... i ....
1 0
310 _305 300 -5
:
:
i 0
i 5
: I 15
I 10
: 45
: I 50
I
I
I
: _ .....
: .'
.. 75
A
,_ 25 E _2o
"
I
I
: :
_ ":....
'. :
-
!
/
.__ 15
g_lo m 'F-
.... 0 -5
'
I
: _ :
: _ :',,LS!--L_
:......
: _ "]1-
Engine Number 1 EngmeNumber2
i" .... !...... !...... ."..... i .....
:
•
. _.-
: ......
:......
;......
] 0
I 5
I 10
: ""_"
:...... 1 15
...... ...... ; ..... I 20
; ...... I 25
'...... I 30
'....... I 35
:...... I 40
; ..... I 45
:....... I 50
;...... I 55
:...... I 60
1...-:-. ,._...._ _ I 65
I 70
75
Time (s)
Figure
6.7:
Simultaneous
Roll
Rate
and
Pitch
5f=0deg).
45
Rate
Doublets
(V=180kts,
H=2,000ft,
--_
0.015,_
,
,
,
,
',
,
,
,
ooi L .... i...... i...... i...... -..... i..... i....: _"
/
" 0.0051-
o I---_" I
rr t'__ "5
:
:
:
:
....:...... :-__-:_-=-_;,--;
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Time (s)
Figure
6.8:
Simultaneous
Roll
Rate
and
Pitch
5F=20deg).
46
(V=180kts,
H=2,000ft,
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Time (s)
Figure 6.9: 5F =0deg).
Simultaneous
Roll Rate
and
Pitch
47
Rate
Doublets
(V=285kts,
H=30,000ft,
X 10 -3 5
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Time (s)
Figure
6.12:
Outer-Loop
Flight
Path
Angle
Step Response
5O
(V=180kts,
H=2,000ft,
6F=0deg).
0.06
i
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