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1. February 24, 2005 1
AAE 666 - Final Presentation
Backstepping Based Flight Control
Asif Hossain

2. February 24, 2005 2
Overview
 Modern Aircraft Configuration
 Aircraft Dynamics
 Force, Moment and Attitude Equations
 Current Approaches to Flight Control Design
 Backstepping Approaches to Flight Control Design
 Backstepping
 Backstepping Design for Flight Control
 Flight Control Laws
 Simulation

3. February 24, 2005 3
Modern Aircraft Configuration

4. February 24, 2005 4
Aircraft Dynamics
, Aircraft position expressed in an
Earth-fixed coordinate system;
, The velocity vector expressed in
the body-axis coordinate system;
, The Euler angles describing the
orientation of the aircraft relative to the Earth-fixed
coordinate system;
, The angular velocity of the
aircraft expressed in the body axes coordinate
system;
T
E
N h
p
p )
(

P
T
w
v
u )
(

V
T
)
( 



Φ
T
r
q
p )
(

ω

5. February 24, 2005 5
Aerodynamics Forces and Moments
Body axis coordinate system

6. February 24, 2005 6
Force Equations (Body-axes)
Z
m
g
pv
qu
w
Y
m
g
ru
pw
v
F
X
m
g
qw
rv
u T
1
cos
cos
1
cos
sin
)
(
1
sin





















Rewrite the force equations in terms of T
V
and
, 






cos
sin
sin
cos
cos
T
T
T
V
w
V
v
V
u



2
2
2
arcsin
arctan
w
v
u
V
V
v
u
w
T
T








7. February 24, 2005 7
Force Equations (wind-axes)
 
1
cos
cos
1
mg
F
D
m
V T
T 


 


)
sin
(
cos
1
tan
)
sin
cos
( 2
mg
F
L
mV
r
p
q T
T






 






)
sin
cos
(
1
cos
sin 3
mg
F
Y
mV
r
p T
T




 





)
cos
cos
sin
sin
sin
cos
sin
sin
cos
(cos
)
sin
sin
cos
cos
(cos
)
cos
cos
cos
sin
sin
cos
sin
sin
cos
cos
(
3
2
1


































g
g
g
g
g
g
Where the contributions due to gravity are given by,

8. February 24, 2005 8
Moment and Attitude Equations
 Moment Equations:
 Attitude Equations:
N
c
L
c
q
r
c
p
c
r
Z
F
M
c
r
p
c
pr
c
q
N
c
L
c
q
p
c
r
c
p
TP
T
9
4
2
8
7
2
2
6
5
4
3
2
1
)
(
)
(
)
(
)
(



























cos
cos
sin
sin
cos
)
cos
sin
(
tan
r
q
r
q
r
q
p











9. February 24, 2005 9
Current Approaches to Flight Control Design
 Gain Scheduling
 Divide and conquer approach is tedious since
controller must be designed for each flight envelope.
 Stability is guaranteed only for low angles of attack
and low angular rates.
 Dynamic Inversion (feedback linearization)
 Cancels valuable nonlinear dynamics.
 Relies on precise knowledge of the aerodynamic
coefficients

10. February 24, 2005 10
Backstepping based flight control design
 Constructive (systematic) control design for
nonlinear systems.
 Lyapunov based control design method
 Avoid cancellation of “useful nonlinearities” (unlike
feedback linearization).
 Stability is guaranteed for all angles of attack
(unlike gain scheduling).
 Different flavors: Adaptive, robust and observer
backstepping.

11. February 24, 2005 11
LaSalle-Yoshizawa Theory
The time-invariant system,
Let be a scalar continuously differentiable function of
the state such that
 is positive definite
 is radially unbounded
 a
Then, all solutions satisfy
In addition, if is positive definite, then the
equilibrium is Globally Asymptotically Stable (GAS).
)
(x
f
x 

)
(x
V
)
(x
V
)
(x
V
x
definite.
semi
positive
is
)
(
where
)
(
)
(
)
( x
W
x
W
x
f
V
x
V x 



0
))
(
(
lim 


t
x
W
t
)
(x
W
0

x

12. February 24, 2005 12
Control Lyapunov Function (clf)
The time-invariant system,
A smooth, positive definite, radially unbounded function
is called a control Lyapunov function (clf) for the system if
for all ,
Given a clf for the system, we can thus find a globally stabilizing control
law. In fact, the existence of a globally stabilizing control law is equivalent
to the existence of a clf, and vice versa.
)
,
( u
x
f
x 

)
(x
V
0

x
u
u
x
f
x
V
x
V x some
for
0
)
,
(
)
(
)
( 



13. February 24, 2005 13
Backstepping
Consider the system
Where are state variables and is the
control input.
Assume a virtual control law is known such that 0
is GAS equilibrium of the system.
u
x
f
x





 )
,
(
R
,
Rn

 
x R

u
)
(x
des

 

14. February 24, 2005 14
Backstepping
Let, be a clf for the subsystem such that
Then, the clf for the augmented system is given by
Moreover, a globally stabilizing control law, satisfying
is given by
)
(x
W )
,
( 
x
f
x 

0
,
0
))
(
,
(
)
(
| 



x
x
x
f
x
W
W des
x
des 



2
))
(
(
2
1
)
(
)
,
( x
x
W
x
V des


 


2
))
(
(
))
(
,
(
)
( x
x
x
f
x
W
V des
des
x 

 


















 )
(
)
(
))
(
,
(
)
,
(
)
(
)
,
(
)
(
x
x
x
x
f
x
f
x
W
x
f
x
x
u des
des
des
x
des

15. February 24, 2005 15
Strict Feedback System
)
,
,
,
(
)
,
,
,
(
)
,
,
(
)
,
(
,
1
1
,
1
2
1
1
1
1
u
x
g
x
g
x
g
x
f
x
m
m
m
i
i
i
i
























By recursive applying backstepping, globally stabilizing
control laws can be constructed for systems of the following
lower triangular form:

16. February 24, 2005 16
Backstepping design for flight control
Controlled variables: General maneuvering

17. February 24, 2005 17
Control Objectives

18. February 24, 2005 18
Assumptions:
 Control surface deflections only produce
aerodynamic moments, and not forces.
 The speed, altitude and orientation of the aircraft
vary slowly compared to the controlled variables.
Therefore, their time derivatives can be neglected.
 Longitudinal and lateral commands are assumed
not to be applied simultaneously.
 The control surface actuator dynamics are assumed
to be fast enough to be disregarded.

19. February 24, 2005 19
Backstepping design for flight control
The roll rate to be controlled, , is expressed in the stability
axes coordinate system.
The stability axes angular velocity, ,is related to
the body axes angular velocity, , through the
transformation:
where
Note that the transformation matrix
Introducing:
T
s
s
s
s r
q
p )
(

ω
T
r
q
p )
(

ω

 sb
s R
















cos
0
sin
-
0
1
0
in
0
cos s
Rsb
T
sb
sb
sb R
R
R 
1
satisfies
s
p
s
T
u
u
u
u 


 )
( 3
2
1

20. February 24, 2005 20
Aircraft Dynamics Revisited
1
u
ps 

)
sin
)
(
(
cos
1
tan 2
mg
F
L
mV
p
q T
T
s
s 




 





2
u
qs 

)
sin
cos
)
(
(
1
3
mg
F
Y
mV
r T
T
s 



 




3
u
rs 

1
2
3
4
5
Roll rate dynamics: Equation 1
Angle of attack dynamics: Equation 2-3
Sideslip dynamics: Equation 4-5

21. February 24, 2005 21
The nonlinear control problem
The angle of attack dynamics and the sideslip dynamics can
be written as
For notational convenience it is favorable to make the
origin the desired equilibrium. Let, is the desired
equilibrium.
u
w
w
y
w
f
w



2
2
1
1 )
,
(


)
,
(
)
,
(
)
(
)
,
(
1
1
2
2
1
1
y
f
y
x
f
x
y
f
w
x
w
x














1
w

22. February 24, 2005 22
The nonlinear control problem
The dynamics become
We will use backstepping to construct a globally stabilizing
feedback control laws for the system assuming general
nonlinearity .
u
x
x
x
x




2
2
1
1 )
(




23. February 24, 2005 23
The nonlinear control problem
Assume there exists a constant, , such that
Then a globally stabilizing control law can be given by
where,
and

0
all
for
)
(
1
1
1



x
x
x

))
(
( 1
2 x
x
k
u 



0
,
0
))
(
)
(
( 1
1
1
1 



 x
x
x
x k
x 

 )
(
0 1

24. February 24, 2005 24
Block Diagram
The nonlinear system is globally stabilized through a cascaded control structure

25. February 24, 2005 25
Aircraft Application
)
sin
)
(
(
cos
1
tan
)
,
( 2
mg
F
L
mV
p
y
f T
T
s 




 



 

)
sin
cos
)
(
(
1
)
,
( 3
mg
F
Y
mV
y
f T
T


 


 


26. February 24, 2005 26
Flight Control Laws
 Angle of attack control
 Sideslip regulation
 Stability axis roll control
))
,
(
)
(
( 1
,
2
,
2 


 

 y
f
k
q
k
u ref
ref
s 




)
sin
cos
1
( 1
,
2
,
3 



 g
V
k
r
k
u
T
s 




)
(
1 s
ref
s
p p
p
k
u s



27. February 24, 2005 27
Gain Selection,
How should the control law parameters be selected?
 For control, linearize the angle of attack dynamics
around a suitable operating point and then select to
achieve some desired linear closed behavior locally
around the operating point.
 For regulation, can be selected by choosing some
desired closed loop behavior using linearization of the
sideslip dynamics.
2
,
1
,
2
,
1
, ,
,
, 


 k
and
k
k
k
k s
p



28. February 24, 2005 28
Roll rate demand, sec
/
1500

ref
s
p

29. February 24, 2005 29
Angle of attack demand, 0
15

ref


30. February 24, 2005 30
Questions
Polygonia interrogationis known as Question Mark