Development of An Intelligent Flight Propulsion Control System

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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



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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