Output-Feedback Control of the Longitudinal Flight Dynamics Using Adaptative Backstepping

Output-Feedback Control of the Longitudinal Flight Dynamics Using
Adaptative Backstepping
F.Gavilan
y
R.Vazquez
y
J.

A.Acosta
?
y
Dpt.de Ingeniera Aeroespacial
?
Dpt.de Ingeniera de Sistemas y Automatica
Universidad de Sevilla
SPAIN
50th IEEE CDC & ECC,Orlando 2011
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 1/18
Outline
1
Introduction
2
Problem statement
3
Controllers Design
Aerodynamic velocity controller
Flight-path-angle controller
4
Simulations
5
Conclusions & Future Work
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 2/18
Outline
1
Introduction
2
Problem statement
3
Controllers Design
Aerodynamic velocity controller
Flight-path-angle controller
4
Simulations
5
Conclusions & Future Work
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 3/18
Introduction
Airplane ight control system for any operating point (Theory)
Minimum aerodynamic knowledge (Theory)
To y an actual aircraft (Application)
Goal:
Nonlinear equations
Aerodynamic models are dicult to obtain and often inaccurate
Measured states )outputs
Awkwardness:
Consider only airplane longitudinal motion
Controllers:the aerodynamic velocity + ight path angle
Adaptive Backstepping
++
scheme
Proposed approach:
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 4/18
Outline
1
Introduction
2
Problem statement
3
Controllers Design
Aerodynamic velocity controller
Flight-path-angle controller
4
Simulations
5
Conclusions & Future Work
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 5/18
Aircraft model
Longitudinal aircraft equations of
motion in wind axes
_
V
a
=
1
m
(D+F
T
cos  mg sin )
_ =
1
mV
a
(L+F
T
sin mg cos )
_
 = q
_q =
M(
e
)
I
y
 =  +
m is the aircraft mass
I
y
is the aircraft inertia
V
a
is the aerodynamic velocity
, and  are the attack, ight-path and pitch angles
q is the pitch velocity
D,L and M are the drag,lift and aerodynamic pitching moments
F
T
is the thrust
With:
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Aircraft model
Longitudinal aircraft equations of
motion in wind axes
_
V
a
=
1
m
(D+F
T
cos  mg sin )
_ =
1
mV
a
(L+F
T
sin mg cos )
_
 = q
_q =
M(
e
)
I
y
 =  +
Aerodynamic velocity V
a
Flight path angle
Control problem
Thrust F
T
.
Elevator de ection 
e
,through M = f(
e
)
Control inputs
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
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Aerodynamic model
Aerodynamic coecients denition:
L =
1
2
V
2
a
SC
L
;D =
1
2
V
2
a
SC
D
;M =
1
2
V
2
a
ScC
m
X We need a model for C
L
,C
D
and C
m
valid in all the ight envelope
Aerodynamic model used:
C
L
= f()
C
D
= C
D
0
+k
1
 +k
2

2
C
m
= C
m
0
+C
m

 +C
m
q
q +C
m

e

e
The lift coecient satises   C
L
()  0
Parabolic drag model satises C
D
> 0
The aerodynamic moment coecient satises C
m

e
< 0
Aerodynamic model of most conventional airplanes (\normal" ight conditions)
ETSI,University of Seville (SPAIN)
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Control problem statement
Improve the previous design of the ight-path-angle controller such that:
NO knowledge of C
L
,C
D
and C
m
Drop assumption on trim angle of attack
Change full-state feedback by output feedback
(Global) regulation in any ight condition
Objectives
Two independent controllers (for now):
Aerodynamic velocity controller:
Thrust (F
T
) as control signal
Drag model coecients unknown
Adaptive control
Measured variables:(V
a
; ;;q)
Flight-path-angle controller:
Elevator (
e
) as control input
ALL aerodynamic moment
coecients unknown
Lift shape known
Adaptive Backstepping approach
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
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Outline
1
Introduction
2
Problem statement
3
Controllers Design
Aerodynamic velocity controller
Flight-path-angle controller
4
Simulations
5
Conclusions & Future Work
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 9/18
Aerodynamic velocity controller
System:
_
V
a
=
1
m


1
2
V
2
a
SC
D
+F
T
cos  mg sin

_z
V
= 
1

z
2
V
+V
2
ref
+2z
V
V
ref

'()
T
 
V
+F
T
cos 
m
g sin +
_
V
ref
where
z
V
= V
a
V
ref
;'() =

1  
2

T
;
V
=

C
D
0
k
1
k
2

T
;
1
=
S
2m
Adaptive-state feedback law:
F
T
=
m
cos 

g sin +
_
V
r
+
1
(z
2
V
+V
2
r
)'()
T

^

V

V
1
z
V

_
^

V
= 
1
z
V

z
2
V
+V
2
r


V
'
V
()
Lyapunov function:
W
V
=
1
2
z
2
V
+
1
2
~

T
V

V
1
~

V
Proposition:global boundedness of z
V
and
^
,and convergence of z
V
to zero.
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Flight-path-angle controller I
System:
_ =
1
mV
a
(L+F
T
sin mg cos )
_
 = q
_q = M(
e
)=I
y
9
=
;
!
_ = f() = f(  )
_
 = q
_q =
V
2
a
S
2I
y
(C
m
0
+C
m


+C
m
q
q +C
m

e

e

Assumption:cos  cos
ref
Property:( 
0
)f()  0;

0
:trim angle of attack,i.e.f(
0
) = 0
f()/
1
2
V
2
a
SC
L
()+F
T
sinmg cos
ref
f() unknown )
0
is NOT computable
Assumptions for the control design:
_
ref
= 0 )regulation
F
T
 0
Error coordinates:
z
1
= 
ref
z
2
= 
ref

0
z
3
= q
#
(x):= f(x+
0
);x(x)  0;x 2 R
_z
1
= (z
2
z
1
)
_z
2
= z
3
_z
3
= 
2
(C
m
0
+C
m

(z
2
z
1
+
0
)
+C
m
q
z
3
+C
m

e

e

ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
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Flight-path-angle controller II
To make the origin z = 0 globally asymptotically stable through the input (
e
).
Notice that z = 0,( ;;q) = (
ref
;
ref
;0)
Control objective
Adaptive Backstepping
++
scheme
C1.
ref
is a given reference.
C2.
0
is unknown and so,
ref
:= 
0
+
ref
too.
C3.C
m

e
is unknown but assumed to be negative
C4.The measurable output vector y 2 R
3
is dened as
y:=
2
4

ref

q
3
5

2
4
z
1
z
2
z
1
+
0
z
3
3
5
Remind::R 7!R unknown,but with the property x (x)  0 (Krstic'95,Harkegard'03)
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Flight-path-angle controller III
...after three steps yields
_z
1
= ;
_
z
}|
{
z
2
k

1
z
1
= (z
3
+c
1
z
1
) c
1
z
1
+

1
;
_z
}|
{
z
3
+c
1
z
1
= 
2
C
m

e

'
T

 

+
e


2


3
(z
3
+c
1
z
1
) +c
1
;
with 
2
=
V
2
a
Sc
2I
y
,

e
= 
2
C
m

e
'

:=
2
6
6
4
1
y
2
y
3


3
(y
3
+c
1
y
1
)
3
7
7
5
=
2
6
6
4
1

q


3
(q +c
1
( 
ref
))
3
7
7
5
;

=
1
C
m

e
2
6
6
4
Cm
0
C
m

C
m
q
1
3
7
7
5
Lyapunov function:
W
3
=
c
1
2
z
2
1
+
Z
z
2
z
1
0
(s)ds +
c
3
2
(z
3
+c
1
z
1
)
2
+
jC
m

e
j
2
~

T


1

~


;
Prop.:The equilibrium manifold ( ;;q;
^


) = (
ref
;
ref
;0;
^



) is GAS (
^



constant),
with the adaptive output-feedback given by

e
= '

(y)
T

^


;
_
^


= 

2
c
1
(q +c
1
( 
ref
)) 

'

(y)
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 13/18
Outline
1
Introduction
2
Problem statement
3
Controllers Design
Aerodynamic velocity controller
Flight-path-angle controller
4
Simulations
5
Conclusions & Future Work
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 14/18
Simulations
Aerodynamic velocity & Cero
0
50
100
150
200
250
70
75
80
85
90
95
100
105
110
Time [s]
Velocity [km/h]


V
V
r
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 15/18
Simulations
Flight-path angle & control inputs
Saturations in control signals included.
F
T
2 [4:9 N;117:6 N]

e
2 [30
o
;30
o
]
0
50
100
150
200
250
−4
−2
0
2
4
6
8
10
12
14
Time [s]
γ [deg]


γ
γ
ref
0
50
100
150
200
250
−40
−20
0
20
40
Time [s]
δe [deg]
0
50
100
150
200
250
−20
0
20
40
60
80
Time [s]
FT [N]
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 15/18
Simulations
States & estimated parameters
0
50
100
150
200
250
0
5
10
Time [s]
α [deg]
0
50
100
150
200
250
−10
0
10
20
Time [s]
θ [deg]
0
50
100
150
200
250
−10
−5
0
5
Time [s]
q [deg/s]
0
50
100
150
200
250
−2000
−1500
−1000
−500
0
500
1000
Time [s]
Estimated parameters


C
m0
/C
mδ
e
C
mα
/C
mδ
e
C
mq
/C
mδ
e
1/C
mδ
e
C
D0
k
1
k
2
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 15/18
Outline
1
Introduction
2
Problem statement
3
Controllers Design
Aerodynamic velocity controller
Flight-path-angle controller
4
Simulations
5
Conclusions & Future Work
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 16/18
Conclusions & Future Work
A controller for a nonlinear longitudinal-aircraft dynamics is proposed
Controller valid for any operating point
NO knowledge of the aerodynamic model required )independent of the airplane
Drop 
0
by output feedback )independent of the lift curve
Explicit and easy-to-implement control law
General Conclusions
Output feedback with the cost of losing Exponential stability (IP)

ref
(t) )tracking.Proof?
Related work:extremum-seeking?
Include a propulsive model to use the throttle as control input
Global stability of the equilibrium of the whole system
Include constraints and saturations for a real ight test
Technical comments & Future Work
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 17/18
...to be continued
Thanks!
ETSI,University of Seville (SPAIN)
Output-Feedback Control of UAVs
IEEE CDC & ECC,Orlando 2011 18/18