Constructive Non-linear Control Design With

Applications to Quad-Rotor and Fixed-Wing

Aircraft

Nicholas Kottenstette, Joseph Porter

ISIS-Vanderbilt University

December 2, 2009

2

Outline

To make (distributed) control system design and synthesis

intuitive and robust via constructive formalisms

Recent Milestones

Fixed-Wing Aircraft Literature Review

Main Idea (back-stepping control)

Quad-rotor Dynamics/ Control/ Simulation

Fixed-Wing Aircraft Dynamics

Airframes Library

Fixed-Wing Control Equations

Velocity Control Subsystem

Body Angular Velocity Control

Angle-of-attack, slide-slip-angle, bank-angle Control

Preliminary Simulations

Future Directions

3

Recent MilestonesJournal Papers:

Kottenstette N., Hall J., Koutsoukos X., Antsaklis P., Sztipanovits J., "Digital

Control of Multiple Discrete Passive Plants Over Networks", Int. J. of

Systems, Control and Communications Special Issue on “Progress in

Networked Control Systems”, pp. 1-20 (to appear)

Conference / Workshop Papers:

N. Kottenstette, J. Porter “Digital Passive Attitude and Altitude Control

Schemes for Quadrotor Aircraft”, 7th IEEE International Conference on

Control & Automation (ICCA’09), Christchurch, New Zealand, December

2009. (invited chair Aircraft Control and Aerodynamic Systems)

Kottenstette, N., Chopra, N., "Lm2-stable digital-control networks for

multiple continuous passive plants", NecSys'09, Venice, Italy, September,

2009.

Nicholas Kottenstette, Gabor Karsai, and Janos Sztipanovits, “A Passivity-

based Framework for Resilient Cyber-Physical Systems”, 2nd International

Symposium on Resilient Control Systems (ISRCS 2009), August 11-13,

2009, Idaho Falls, Idaho.

Technical Report:

Kottenstette, N., H. LeBlanc, E. Eyisi, and X. Koutsoukos, "Multi-Rate

Networked Control of Conic Systems", Technical Report, Nashville, TN,

Institute for Software Integrated Systems, Vanderbilt University, 09/2009.

4

Fixed-Wing Aircraft Literature Review

Key Fixed-Wing Aircraft Control:

J. Farrell, M. Sharma, and M. Polycarpou, “Backstepping-based flight

control with adaptive function approximation,” Journal of Guidance, Control,

and Dynamics, vol. 28, no. 6, pp. 1089–1102, 2005.

O. Harkegard, “Backstepping and control allocation with applications to

flight control,” Ph.D. dissertation, Linkoping University, 2003.

B. L. Stevens and F. L. Lewis, Aircraft Control and Simulation, 2nd ed. John

Wiley & Sons, Inc., 2003.

S.L. Morris, D.E. Bossert and W.F. Hallgren, Introduction to Aircraft Flight

Mechanics, Performance, Static Stability, Dynamic Stability, and Classical

Feedback Control, AIAA Education Series, 2003.

Key Fixed-Wing Aircraft Formation Control:

G. Campa, Y. Gu, B. Seanor, M. Napolitano, L. Pollini, and M. Fravolini,

“Design and flight-testing of non-linear formation control laws,” Control

Engineering Practice, vol. 15, no. 9, pp. 1077–1092, 2007. [airlib]

Y. Zou, P. Pagilla, and R. Ratliff, “Distributed Formation Flight Control Using

Constraint Forces,” JOURNAL OF GUIDANCE, CONTROL, AND

DYNAMICS, vol. 32, no. 1, 2009.

5

Main Idea (back-stepping control)

1( )f x

1( )g x1x

1x

2 ( )f x

2 ( )g x2x

2x1u

1 1 1 1( ) ( )x f x g x u

2 2 2 1( ) ( )x f x g x x

J. Farrell, M. Sharma, and M. Polycarpou, “Backstepping-based flight control with

adaptive function approximation,” Journal of Guidance, Control, and Dynamics, vol. 28,

no. 6, pp. 1089–1102, 2005.

6

Recursive Control Strategy

1( )f x

1( )g x1x

1x1u1

1 ( )g x

1( )f x

1k1dx

1dx

111 1 1 1 1 1( ) ( ) ( ) ddu g x k x x f x x

7

Ideal Feedback and Feedforward Comp.

1x

1x

1k1dx

1dx

11dx 1x

8

Recursive Control Strategy

2 ( )f x

2 ( )g x2x

2x1dx1

2 ( )g x

2 ( )f x

2k2dx

2dx

121 2 2 2 2 2( ) ( ) ( ) dd dx g x k x x f x x

9

Quad-Rotor Dynamics

( )

( )

( )

0 1 sin( ) tan( ) cos( ) tan( )

[ , , ] , 0 , ( ) 0 cos( ) sin( )

0 sin( ) cos( )0

cos( ) cos( )

[ , , ] , ( )

I

TI I D ab

T

T

v

mv f mge R f

I I

J

r q

p q r r p J

q p

c c c s s

R s s

c s c

( ) ( ) , ( ) ( ) ( ), ( )T T

I

c c s s s s c c c s

s s c s s s c c c

R R I R R R

10

Quad-Rotor Control

11

Quad-Rotor Simulation

0 10 20 30 40 50 60 70 80-25

-20

-15

-10

-5

0

5

10

15

20

25

time (s)

(m

)

xI

yI

zI

12

Quad-Rotor Simulation

0 10 20 30 40 50 60 70 80

-3

-2

-1

0

1

2

3

time (s)

(ra

dia

ns)

13

Fixed-Wing Aircraft Dynamics

2 1 1

/

( ) , diag , , , ,

1[ , , ] , , , tan , sin

2

cos( ) 0 sin( )

[ , , ] , ( ) 0 1 0

sin( ) 0 cos( )

T

l m n d

T T T

b b b I I

Ds

T

as Ds Y L ab b s as Y

L

I I qS b c b C C C

w vv u v w V v v v v q V

u V

C

f qS C C C f R f qS C

C

DDin Dq

Yp Yr Y Y

Lin Lq L

0 00 01

0 02

0 0 0 0

e

a r

e

Ds D Do f e

Y Y Yo a

L L Lo f r

l l

m

n

CC C C C C c p

C C C C b C b q C CV

C C C C C c r C

C C

C

C

l llp lr

min mq m

np nr m m

001

0 0 0 02

0 0

a r

e

a r

lo e

m mo f a

n no r

C CC C b C b p

C C C C c q CV

C C C b C b r C C

14

Airframes Library

http://www.ae.illinois.edu/m-selig/apasim/Aircraft-uiuc.html

15

Airframes Library

16

Relevant Fixed-Wing Equations for Control

1

2

3

cos cos

cos sin

cos sin cos sin( )

cos 0 sin cos sin 0

0 1 0 , sin cos 0

sin 0 cos 0 0 1

T

T s

T s

s

s

s

mV F D mg

mV F C mg mVr

mV F L mg mV q p

p p D

q q C qS

r r L

1 3

1 11 2

2 2 2 2

1 2 1 2

1 1

Velocity flight-path angle, , - =sin

Velocity heading-angle, cos sin

Bank-angle, cos sincos

Ds

Y

L

I

I I

I I I I

C

C

C

v

V

v v

v v v v

c s c cc c c s s

1 2 3

cos

( )sin , ( )sin cos , ( ) cos cos

s s c c s

g g h g g h g g h

17

Relevant Fixed-Wing Equations for Control

1cos sin cos

1cos sin

1

cos

T

T

sT

gC L F s c c s s

V mV

C L F s s c s cmV

pgCt c L t t s F s s t s t c s c t t c c

mV V

Velocity

ControllerdV TF

Vel-Angle

Controller

,T

d d ,T

d d

0

Aerodynamic

-Angle

Controller

, ,T

d d dp q r

Body-

angular

Velocity

Controller

, ,T

e a r

18

Velocity Control Subsystem

Do Din

Dq

D D

Do Din set

( )2

( ) ( )

e

T f v

v e

dT v f d I

mV F qS C C f x

C cf x qS C q C mg h

V

sF f x mg h qS C C m V mK V V

s

set

( )

( )

I

I

KV s

V s s K

19

Angle-of-attack, slide-slip-angle, bank-angle Control

1 00 0 0( )

0 0 ( ) 0 1 04

( )

0 0 0 14 4

( )

LqT L

T YYp Yr

a

f x pC cF qSC

f x qmV m

f x rF qSC C b C b

mV m m

a A x a

1

2( ) ( ) , ( ) ( ) ( )c a df x G x G x f x k a a

20

Body Angular Velocity Control

set( ) , ( )d dI I k

21

Preliminary Simulations

22

Future Directions

Short Term:

Finish Flight-Path Heading Angle and Velocity Heading Angle

Controller Design

Evaluate Controller Performance When Subject to Wind

Disturbances

Include 101-[1,4] and 201-[1,4] airframe-configurations

described in WL-TR-97-3059

Long Term:

Use Power-Junction Like Structure to Implement Formation

Control System

Investigate Adaptive Control Structures

Consider Actuator Rate and Dynamic Limitations

Improve back-stepping control synthesis tools