Longitudinal Dynamics and Adaptive Control Application for an Aeroelastic Generic Transport Model

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Nov 16, 2013 (3 years and 11 months ago)

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Longitudinal Dynamics and Adaptive Control Application for
an Aeroelastic Generic Transport Model
Nhan Nguyen

NASA Ames Research Center,Moffett Field,CA 94035
Ilhan Tuzcu

California State University,Sacramento,CA 95819
Tansel Yucelen

and Anthony Calise
§
Georgia Institute of Technology,Atlanta,GA 30332
This paper presents an aeroelastic model of longitudinal dynamics of a generic transport model (GTM).
Aeroelasticity theory is used to develop an aeroelastic flight dynamic model of the flexible GTMto account for
interactions between wing bending and torsion on aircraft performance and stability.The Galerkin’s method is
used to implement a weak-formsolution of the aeroelastic equations of the aircraft.The weak-formaeroelastic
equations are then coupled with the longitudinal dynamic equations of the rigid-body aircraft to formulate
an aeroelastic flight dynamic model.This model is then used to create a reduced-order state space model of
the rigid-body longitudinal dynamics with the flexible-body dynamics represented as unmodeled dynamics.
Matched uncertainty and wind gust disturbances are introduced in the model and is effectively addressed
by two recently developed robust modification adaptive control methods:Optimal Control Modification and
Adaptive Loop Recovery.Both methods demonstrate the effectiveness in reducing the effects of the uncertainty
and wind gust disturbances.
I.Introduction
Light weight aircraft design has received a considerable attention in recent years as a means for improving cruise
efficiency.Reducing aircraft weight results in lower lift requirement which directly translates into lower induced drag,
hence reduced engine thrust requirement during cruise.The use of light-weight materials such as advanced composite
materials has been adopted by airframe manufacturers in a number of current and future aircraft.Modern light-
weight materials can provide less structural rigidity while maintaining sufficient load-carrying capacity.As structural
flexibility increases,aeroelastic interactions with aerodynamic forces and moments become an increasingly important
consideration in aircraft design.Understanding aeroelastic effects can improve the prediction of aircraft aerodynamic
performance and provide an insight into how to design an aerodynamically efficient airframe that exhibits a high
degree of flexibility.Moreover,structural flexibility of airframes can cause significant aeroelastic interactions that can
degrade vehicle stability margins,potentially leading to loss of control.There exists a trade-off between the desire
of having light weight,flexible structures for weight savings and the need for maintaining sufficient robust stability
margins fromaeroelastic instability.
This paper describes an aeroelastic model of a generic transport model (GTM).The aeroelastic model is based
on one-dimensional structural dynamic theory that models a wing structure as a one-dimensional elastic member
in a combined coupled bending-torsion motion.Aeroelastic analysis is performed based on the quasi-steady state
aerodynamic assumption.Flight control simulations of aircraft response to gust loads are performed.Two adaptive
control schemes based on the optimal control modification
1
and adaptive loop recovery
2
are designed as adaptive
augmentation controllers to demonstrate the effectiveness of gust load alleviation and uncertainty accommodation
using adaptive control.

Research Scientist,Associate Fellow AIAA,Intelligent Systems Division,nhan.t.nguyen@nasa.gov

Assistant Professor,AIAA Senior Member,Mechanical Engineering Department,tuzcui@ecs.csus.edu

Graduate Research Assistant,Student Member AIAA,School of Aerospace Engineering,tansel@gatech.edu
§
Professor,Fellow Member AIAA,School of Aerospace Engineering,anthony.calise@aerospace.gatech.edu
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American Institute of Aeronautics and Astronautics
II.Aeroelastic Flight Dynamic Modeling
The aeroelastic effects of the generic transport model (GTM) is assumed to be contributed only by the wing
structures.Fuselage and tail surface deflections are assumed to be negligible.For this study,aeroelasticity theory is
used to develop an aeroelastic flight dynamic model of the GTMto account for interactions between wing bending and
torsion on aircraft performance and stability only in the pitch axis.Future work will extend the analysis to the other
axes.
A.Reference Frames
Figure 1 - Aircraft Reference Frames
Figure 1 illustrates three orthogonal views of a typical aircraft.Several reference frames are introduced to facilitate
the rigid-body dynamic and structural dynamic analysis of the lifting surfaces.For example,the aircraft inertial
reference frame A is defined by unit vectors a
1
,a
2
,and a
3
fixed to the non-rotating earth.The aircraft body-fixed
reference frame B is defined by unit vectors b
1
,b
2
,and b
3
.The reference frames A and B are related by three
successive rotations:1) the first rotation about a
3
by the heading angle y that results in an intermediate reference
frame A
0
defined by unit vectors a
0
1
,a
0
2
,and a
0
3
(not shown),2) the second rotation about a
0
2
by the pitch angle q
that results in an intermediate reference frame B
0
defined by unit vectors b
0
1
,b
0
2
,and b
0
3
(not shown),and 3) the third
rotation about b
0
1
by the bank angle f that results in the reference frame B.This relationship can be expressed as
2
6
4
a
1
a
2
a
3
3
7
5=
2
6
4
cosy siny 0
siny cosy 0
0 0 1
3
7
5
2
6
4
cosq 0 sinq
0 1 0
sinq 0 cosq
3
7
5
2
6
4
1 0 0
0 cosf sinf
0 sinf cosf
3
7
5
2
6
4
b
1
b
2
b
3
3
7
5
=
2
6
4
cosycosq sinycosf +cosysinq sinf sinysinf +cosysinq cosf
sinycosq cosycosf +sinysinq sinf cosysinf +sinysinq cosf
sinq cosq sinf cosq cosf
3
7
5
2
6
4
b
1
b
2
b
3
3
7
5
(1)
The left wing elastic reference frame D is defined by unit vectors d
1
,d
2
,and d
3
.The reference frames B and
D are related by three successive rotations:1) the first rotation about b
3
by the elastic axis sweep angle
3p
2
 that
results in an intermediate reference frame B

defined by unit vectors b

1
,b

2
,and b

3
(not shown),2) the second rotation
about negative b

2
by the elastic axis dihedral angle  that results in an intermediate reference frame D
0
defined by unit
vectors d
0
1
,d
0
2
,and d
0
3
(not shown),and 3) the third rotation about d
0
1
by an angle p that results in the reference frame
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American Institute of Aeronautics and Astronautics
D.This relationship can be expressed as
2
6
4
b
1
b
2
b
3
3
7
5
=
2
6
4
sin cos 0
cos sin 0
0 0 1
3
7
5
2
6
4
cos 0 sin
0 1 0
sin 0 cos
3
7
5
2
6
4
1 0 0
0 1 0
0 0 1
3
7
5
2
6
4
d
1
d
2
d
3
3
7
5
=
2
6
4
sincos cos sinsin
coscos sin cossin
sin 0 cos
3
7
5
2
6
4
d
1
d
2
d
3
3
7
5
(2)
Generally,the effect of the dihedral angle can be significant.A full analysis with the dihedral angle can be
performed but can also result in a very complex analytical formulation.Thus,to simplify the analysis,the dihedral
effect is assumed to be negligible in this study.The right wing reference frame C can be established in a similar
manner.In the analysis,the aeroelastic effects on the fuselage,horizontal stabilizers,and vertical stabilizer are not
considered,but the analytical method can be formulated for analyzing these lifting surfaces if necessary.In general,a
whole aircraft analysis approach should be conducted to provide a comprehensive assessment of the effect of structural
flexibility on aircraft performance and stability.However,the scope of this study pertains to only the wing structures.
B.Elastic Analysis
In the subsequent analysis,the combined motion of the left wing is considered.The motion of the right wing is a
mirror image of that of the left wing for symmetric flight.The wing has a varying pre-twist angle g (x) commonly
designed in many aircraft.Typically,the wing pre-twist angle varies frombeing nose-up at the wing root to nose-down
at the wing tip.The nose-down pre-twist at the wing tip is designed to delay stall onsets.This is called a wash-out
twist distribution.Under aerodynamic forces and moments,the aeroelastic deflections of a wing introduce stresses
and strains into the wing structure.The internal structure of a wing typically comprises a complex arrangement of load
carrying spars and wing boxes.Nonetheless,the elastic behavior of a wing can be captured by the use of equivalent
stiffness properties.These properties can be derived from structural certification testing that yields information about
wing deflections as a function of loading.For high aspect ratio wings,an equivalent one-dimensional elastic approach
can be used to analyze aeroelastic deflections with good accuracy.The equivalent one-dimensional elastic approach is
a typical formulation in many aeroelasticity studies.
3
It is assumed that the effect of wing curvature is ignored and the
one-dimensional aeroelasticity theory is used to model the wing aeroelastic deflections.
Consider an airfoil section on the left wing as shown in Figure 2 undergoing bending and twist deflections.
Figure 2 - Left Wing Reference Frame
Let (x;y;z) be the coordinates of a point Q on the airfoil.Then the undeformed local airfoil coordinates of point Q
are
"
y
z
#
=
"
cosg sing
sing cosg
#"
h
x
#
(3)
where h and x are local airfoil coordinates,and g is the wing section pre-twist angle,positive nose-down.
4
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American Institute of Aeronautics and Astronautics
Then differentiating with respect to x gives
"
y
x
z
x
#
=g
0
"
sing cosg
cosg sing
#"
h
x
#
=
"
zg
0
yg
0
#
(4)
The axial or extensional deflection of a wing is generally very small and therefore can usually be neglected.Let
be a torsional twist angle about the x-axis,positive nose-down,and let W and V be flapwise and chordwise bending
deflections of point Q,respectively..Then,the rotation angle due to the elastic deformation can be expressed as
f (x;t) =d
1
W
x
d
2
+V
x
d
3
(5)
where the subscripts x and t denote the partial derivatives of ,W,and V.
Let (x
1
;y
1
;z
1
) be the coordinates of point Qon the airfoil in the reference frame D.Then the coordinates (x
1
;y
1
;z
1
)
are computed using the small angle approximation as
2
6
4
x
1
(x;t)
y
1
(x;t)
z
1
(x;t)
3
7
5
=
2
6
4
x
y+V
z +W
3
7
5
+
2
6
4
f (yd
2
+zd
3
):d
1
f (yd
2
+zd
3
):d
2
f (yd
2
+zd
3
):d
3
3
7
5
=
2
6
4
xyV
x
zW
x
y+V z
z +W+y
3
7
5
(6)
Differentiating x
1
,y
1
,and z
1
with respect to x yields
2
6
4
x
1;x
y
1;x
z
1;x
3
7
5
=
2
6
4
1yV
xx
+zg
0
V
x
zW
xx
yg
0
W
x
zg
0
+V
x
z
x
yg
0

yg
0
+W
x
+y
x
zg
0

3
7
5
(7)
Neglecting the transverse shear effect,the longitudinal strain is computed as
5
e =
ds
1
ds
ds
=
s
1;x
s
x
1 (8)
where
s
x
=
q
1+y
2
x
+z
2
x
=
q
1+(y
2
+z
2
)

g
0

2
(9)
s
1;x
=
q
x
2
1;x
+y
2
1;x
+z
2
1;x
=
q
1+(y
2
+z
2
)

g
0

2
2yV
xx
2zW
xx
+2(y
2
+z
2
)g
0

x
(10)
For a small wing twist angle g,the longitudinal strain is obtained as
e =yV
xx
zW
xx
+

y
2
+z
2

g
0

x
(11)
The moments acting on the wing are then obtained as
2
6
4
M
x
M
y
M
z
3
7
5
=
2
6
4
GJ
x
0
0
3
7
5
+

Ee
2
6
4

y
2
+z
2


g
0
+
x

z
y
3
7
5
dydz (12)
=
2
6
6
4
GJ +EB
1

g
0

2
EB
2
g
0
EB
3
g
0
EB
2
g
0
EI
yy
EI
yz
EB
3
g
0
EI
yz
EI
zz
3
7
7
5
2
6
4

x
W
xx
V
xx
3
7
5
(13)
where E is the Young’s modulus;G is the shear modulus;g
0
is the derivative of the wing pre-twist angle;I
yy
,I
yz
,and
I
zz
are the section area moments of inertia about the flapwise axis;J is the torsional constant;and B
1
,B
2
,and B
3
are
the bending-torsion coupling constants which are defined as
2
6
4
B
1
B
2
B
3
3
7
5=


y
2
+z
2

2
6
4
y
2
+z
2
z
y
3
7
5dydz (14)
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American Institute of Aeronautics and Astronautics
The strain analysis shows that for a pre-twisted wing the bending deflections W and V are coupled to the torsional
deflection via the slope of the wing pre-twist angle.This coupling can be significant if the termg
0
is dominant as in
highly twisted wings such as turbomachinery blades.For an aircraft wing structure,a simplification can be made by
neglecting the chordwise bending deflection.Thus,the resulting moments are now given as
"
M
x
M
y
#
=
2
4
GJ +EB
1

g
0

2
EB
2
g
0
EB
2
g
0
EI
yy
3
5
"

x
W
xx
#
(15)
C.Aeroelastic Angle of Attack
The relative velocity of the air approaching a wing section includes the contribution from the wing elastic deflection
that results in changes in the local angle of attack.Since aerodynamic forces and moments are dependent on the local
angle of attack,the wing aeroelastic deflections will generate additional elastic forces and moments.The local angle
of attack depends on the relative approaching air velocity as well as the rotation angle f from Eq.(5).The relative
air velocity in turn also depends on the deflection-induced velocity.The local velocity components at point Q in the
reference frame D are given by
4
2
6
4
v
x
v
y
v
z
3
7
5
=
2
6
4
usin+x
1;t
ucos+y
1;t
wqx
a
+z
1;t
3
7
5
=
2
6
4
usinzW
xt
ucosz
t
wqx
a
+W
t
+y
t
3
7
5
(16)
where u V

,wV

a,q is the aircraft pitch rate,x
a
is the position of point Qwith respect to the aircraft C.G.(positive
aft of C.G.) measured in the aircraft reference frame B,and y and z are coordinates of point Q in the reference frame
D.
In order to compute the aeroelastic forces and moments,the velocity must be transformed fromthe reference frame
D to the airfoil local coordinate reference frame defined by (m;h;x) as shown in Figure 3.Then the transformation
can be performed using two successive rotation matrix multiplication operations as
2
6
4
v
m
v
h
v
x
3
7
5
=
2
6
4
1 0 0
0 cos(+g) sin(+g)
0 sin(+g) cos(+g)
3
7
5
2
6
4
cosW
x
0 sinW
x
0 1 0
sinW
x
0 cosW
x
3
7
5
2
6
4
v
x
v
y
v
z
3
7
5
=
2
6
4
v
x
cosW
x
+v
z
sinW
x
v
x
sinW
x
sin(+g) +v
y
cos(+g) +v
z
cosW
x
sin(+g)
v
x
sinW
x
cos(+g) v
y
sin(+g) +v
z
cosW
x
cos(+g)
3
7
5
(17)
For small deflections,the local velocity components can be simplified as
2
6
4
v
m
v
h
v
x
3
7
5=
2
6
4
v
x
+v
z
W
x
v
y
+v
z
(+g)
v
z
v
x
W
x
v
y
(+g)
3
7
5 (18)
Referring to Figure 7,the local aeroelastic angle of attack on the airfoil section due to the velocity components v
h
and v
x
in the reference frame D is computed as
a
c
=
v
x
v
h
=
¯v
x
+v
x
¯v
h
+v
h
(19)
where
¯v
x
=wqx
a
(20)
¯v
h
=ucos (21)
v
x
=W
t
+y
t
v
x
W
x
v
y
(+g) (22)
v
h
=z
t
+v
z
(+g) (23)
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American Institute of Aeronautics and Astronautics
Then the local aeroelastic angle of attack can be evaluated as
a
c
=
v
x
¯v
h

¯v
x
v
h
¯v
2
h
=
wqx
a
+W
t
+y
t
v
x
W
x
v
y
(+g)
ucos

(wqx
a
)[z
t
+v
z
(+g)]
u
2
cos
2

(24)
Ignoring the nonlinear terms,the expression for the local aeroelastic angle of attack is obtained as
4
a
c
(x;y;z) =
a
cos
+
qx
a
V

cos
g (x) W
x
tan
W
t
+y
t
V

cos

(w+qx
a
)[z
t
+(w+qx
a
)(+g)]
V
2

cos
2

(25)
The terms W
t
and 
t
contribute to aerodynamic damping forces which can be significant for aeroelastic stability.
Figure 3 - Airfoil Local Coordinates
For aeroelastic analysis,the steady state aerodynamic method assumes that the steady state lift circulation oc-
curs at the aerodynamic center of the oscillating airfoil,which may be taken to be the quarter-point.On the other
hand,the unsteady aerodynamic method assumes that the unsteady circulation acts at the 3=4-chord point.
4
Both
the Theodorsen’s method for simple harmonic airfoil motion
6
and Peters’ finite-state method can be used to analyze
unsteady aerodynamics.
3
Based on the steady state aerodynamic assumption,the local angle of attack of an airfoil
section at the elastic axis is evaluated at y =e and z =0.Neglecting the last term,the expression for a
c
is
a
c
(x) =
a
cos
+
qx
ac
V

cos
g (x) W
x
tan
W
t
e
t
V

cos
(26)
where x
ac
is the distance fromaircraft C.G.to the aerodynamic center measured in aircraft reference frame B (positive
aft of C.G.) e is the distance between the aerodynamic center and the elastic axis.
For unsteady aerodynamics,the local angle of attack is evaluated at y =b

1
2
a

a
c
=
a
cos
+
qx
ac
V

cos
g (x) W
x
tan
W
t
+b

1
2
a


t
V

cos
(27)
where b is the half-chord length and 1 a 1 is a parameter such that the elastic axis is located at a distance ab
fromthe mid-chord and a <0 when the elastic axis is forward of the mid-chord.
D.Wing Aeroelasticity
The equilibriumconditions for bending and torsion are expressed as
¶M
x
¶x
=m
x
(28)

2
M
y
¶x
2
= f
z

¶m
y
¶x
(29)
where m
x
is the pitching moment per unit span about the elastic axis,f
z
is the lift force per unit span,and m
y
is the
bending moment per unit span about the flapwise axis of the wing which is assumed to be zero.
The local lift coefficients and pitching moment are given by
c
L
(x) =c
L
0
+c
L
a
a
c
(x) +c
L
d
d (30)
c
m
(x) =c
m
ac
+
e
c

c
L
0
+c
L
a
a
c
(x)

+
m

k=1

c
m
d
k
+
e
c
c
L
d
k

d
k
(31)
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American Institute of Aeronautics and Astronautics
where c
m
ac
is the section pitching moment coefficient about the section at the aerodynamic center,c
L
0
is the section
lift coefficient at zero angle of attack,c
L
a
is the section lift vs.angle of attack curve slope,c is the section chord,d
k
is
the surface deflection of the k-th flap,and c
L
d
k
and c
m
d
k
are the section lift and pitching moment control derivative at
the quarter-chord point due to the k-th flap.
Figure 4 - Airfoil Forces and Moment
Using the sign convention as shown in Figure 4,the lift force and pitching moment per unit span can be expressed
as
f
z
=
"
c
L
0
+c
L
a

a
cos
+
qx
ac
V

cos
g W
x
tan
W
t
e
t
V

cos

+
m

k=1
c
L
d
k
d
k
#
q

cos
2
c
rgArAW
tt
+rAe
cg

tt
(32)
m
x
=
(
c
e
c
m
ac
+
"
c
L
0
+c
L
a

a
cos
+
qx
ac
V

cos
g W
x
tan
W
t
e
t
V

cos

+
m

k=1

c
e
c
m
d
k
+
c
L
d
k

d
k
#)

eq

cos
2
c+rgAe
cg
rI
xx

tt
+rAe
cg
W
tt
(33)
where q

is the dynamic pressure,r is the wing material density including fuel density,A is the cross sectional area
of a wing section,e
cg
is the eccentricity between the center of mass and the elastic axis (positive corresponding to the
center of mass located forward of the elastic axis),I
xx
is the section polar area moment of inertia,and the termcos
2

accounts for the wing sweep angle  as measured fromthe elastic axis.
The bending and torsion aeroelastic equations then become

2
¶x
2

EB
2
g
0

x
+EI
yy
W
xx

=
"
c
L
0
+c
L
a

a
cos
+
qx
ac
V

cos
g W
x
tan
W
t
e
t
V

cos

+
m

k=1
c
L
d
k
d
k
#
q

cos
2
c
rgArAW
tt
+rAe
cg

tt
(34)

¶x

GJ +EB
1

g
0

2


x
EB
2
g
0
W
xx

=
(
c
e
c
m
ac
+
"
c
L
0
+c
L
a

a
cos
+
qx
ac
V

cos
g W
x
tan
W
t
e
t
V

cos

+
m

k=1

c
e
c
m
d
k
+
c
L
d
k

d
k
#)
eq

cos
2
c
rgAe
cg
+rI
xx

tt
rAe
cg
W
tt
(35)
subject to fixed-end symmetric-mode boundary conditions W(0;t) =W
x
(0;t) =W
xx
(L;t) =
d
dx

EIW
xx
(L;t) EB
2
g
0

x
(L;t)

=
0 and (0;t) =
x
(L;t) =0,whereupon the x-coordinate of the wing elastic axis is translated such that the wing root
section is at x =0 and wing tip section is at x =L.
These equations describe the wing bending and torsional deflections due to aerodynamic forces and moments.
Using the Galerkin’s method,
7
the bending and torsional deflections can be approximated as
W(x;t) =
n

j=1
w
j
(t)
j
(x) (36)
7 of 22
American Institute of Aeronautics and Astronautics
(x;t) =
n

j=0
q
j
(t)
j
(x) (37)
where w
j
(t) and q
j
(t) are the generalized coordinates for static bending and torsion,and 
j
(x) and 
j
(x) are the
assumed normalized eigenfunctions of the j-th bending and torsion aeroelastic modes,respectively,j =1;2;:::;n.
The normalized eigenfunctions are given by

j
(x) =cosh(b
j
x) cos(b
j
x) 
cosh(b
j
L) +cos(b
j
L)
sinh(b
j
L) +sin(b
j
L)
[sinh(b
j
x) sin(b
j
x)] (38)

j
(x) =
p
2sin
(2j 1)px
2L
(39)
where b
j
L =1:87510;4:69409;:::is the eigenvalue of the j-th bending mode of a uniform cantilever beam,and the
eigenfunctions 
j
(x) and 
j
(x) satisfy the orthogonal condition

L
0

i
(x)
j
(x)dx =

L
0

i
(x)
j
(x)dx =
8
<
:
L i = j
0 i 6= j
(40)
The weak-formintegral expressions of the dynamic aeroelastic equations are obtained by multiplying the bending
and torsion aeroelastic equations by 
i
(x) and 
i
(x),and then integrating over the wing span.This yields
n

j=1

L
0

i
d
2
dx
2

EB
2
g
0
q
j

0
j
+EI
yy
w
j

00
j

dx =
n

j=1

L
0

i
"
c
L
0
+c
L
a

a
cos
+
qx
ac
V

cos
w
j

0
j
tanq
j

j

˙w
j

j
e
˙
q
j

j
V

cos

+
m

k=1
c
L
d
k
d
k
#
q

cos
2
cdx


L
0

i
rgAdx
n

j=1

L
0

i
rA ¨w
j

j
dx+
n

j=1

L
0

i
rAe
cg
¨
q
j

j
dx (41)
n

j=1

L
0

i
d
dx

GJ +EB
1

g
0

2

q
j

0
j
EB
2
g
0
w
j

00
j

dx =
n

j=1

L
0

i
(
c
e
c
m
ac
+
"
c
L
0
+c
L
a

a
cos
+
qx
ac
V

cos
w
j

0
j
tanq
j

j

˙w
j

j
e
˙
q
j

j
V

cos

+
m

k=1

c
e
c
m
d
k
+
c
L
d
k

d
k
#)

eq

cos
2
cdx

L
0

i
rgAe
cg
dx+
n

j=1

L
0

i
rI
xx
¨
q
j

j
dx
n

j=1

L
0

i
rAe
cg
¨w
j

j
dx (42)
The expressions of the left hand sides can be integrated by parts as

L
0

i
d
2
dx
2

EB
2
g
0
q
j

0
j
+EI
yy
w
j

00
j

dx = 
i
d
dx

EB
2
g
0
q
j

0
j
+EI
yy
w
j

00
j





L
0

0
i

EB
2
g
0
q
j

0
j
+EI
yy
w
j

00
j




L
0
+

L
0

00
i

EB
2
g
0
¯
q
0
j
+EI
yy
w
j

00
j

dx (43)

L
0

i
d
dx

GJ +EB
1

g
0

2

q
j

0
j
EB
2
g
0
w
j

00
j

dx = 
i

GJ +EB
1

g
0

2

q
j

0
j
EB
2
g
0
w
j

00
j




L
0


L
0

0
i

GJ +EB
1

g
0

2

q
j

0
j
EB
2
g
0
w
j

00
j

dx (44)
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American Institute of Aeronautics and Astronautics
Upon enforcing the zero boundary conditions at the two end points,the weak-formdynamic aeroelastic equations
are obtained as
n

j=1

L
0

00
i

EB
2
g
0
q
j

0
j
+EI
yy
w
j

00
j

dx

n

j=1

L
0

i
"
c
L
0
+c
L
a

a
cos
+
qx
ac
V

cos
w
j

0
j
tanq
j

j

˙w
j

j
e
˙
q
j

j
V

cos

+
m

k=1
c
L
d
k
d
k
#
q

cos
2
cdx
+
n

j=1

L
0

i
rA ¨w
j

j
dx
n

j=1

L
0

i
rAe
cg
¨
q
j

j
dx =

L
0

i
rgAdx (45)
n

j=1

L
0

0
i

GJ +EB
1

g
0

2

q
j

0
j
EB
2
g
0
w
j

00
j

dx
+
n

j=1

L
0

i
(
c
e
c
m
ac
+
"
c
L
0
+c
L
a

a
cos
+
qx
ac
V

cos
w
j

0
j
tanq
j

j

˙w
j

j
e
˙
q
j

j
V

cos

+
m

k=1

c
e
c
m
d
k
+
c
L
d
k

d
k
#)

eq

cos
2
cdx+
n

j=1

L
0

i
rI
xx
¨
q
j

j
dx
n

j=1

L
0

i
rAe
cg
¨w
j

j
dx =

L
0

i
rgAe
cg
dx (46)
These equations can be expressed as
n

j=1

m
w
i
q
j
¨
q
j
+m
w
i
w
j
¨w
j
+c
w
i
q
j
˙
q
j
+c
w
i
w
j
˙w
j
+k
w
i
q
j
q
j
+k
w
i
w
j
w
j
+h
w
i
a
a +h
w
i
q
q

= f
w
i
+
m

k=1
g
w
i
d
k
d
k
(47)
n

j=1

m
q
i
q
j
¨
q
j
+m
q
i
w
j
¨w
j
+c
q
i
q
j
˙
q
j
+c
q
i
w
j
˙w
j
+k
q
i
q
j
q
j
+k
q
i
w
j
w
j
+h
q
i
a
a +h
q
i
q
q

= f
q
i
+
m

k=1
g
q
i
d
k
d
k
(48)
where
m
w
i
w
j
=

L
0
rA
i

j
dx (49)
m
w
i
q
j
=

L
0
rAe
cg

i

j
dx (50)
m
q
i
w
j
=

L
0
rAe
cg

i

j
dx (51)
m
q
i
q
j
=

L
0
rI
xx

i

j
dx (52)
c
w
i
w
j
=
1
2
r

V


L
0
c
L
a
cosc
i

j
dx (53)
c
w
i
q
j
=
1
2
r

V


L
0
c
L
a
eccos
i

j
dx (54)
c
q
i
w
j
=
1
2
r

V


L
0
c
L
a
eccos
i

j
dx (55)
c
q
i
q
j
=
1
2
r

V


L
0
c
L
a
e
2
ccos
i

j
dx (56)
k
w
i
w
j
=

L
0
EI
yy

00
i

00
j
dx+q


L
0
c
L
a
ctancos
2

i

0
j
dx (57)
k
w
i
q
j
=

L
0
EB
2
g
0

00
i

0
j
dx+q


L
0
c
L
a
ccos
2

i

j
dx (58)
9 of 22
American Institute of Aeronautics and Astronautics
k
q
i
w
j
=

L
0
EB
2
g
0

0
i

00
j
dxq


L
0
c
L
a
ectancos
2

i

0
j
dx (59)
k
q
i
q
j
=

L
0

GJ +EB
1

g
0

2


0
i

0
j
dxq


L
0
c
L
a
eccos
2

i

j
dx (60)
h
w
i
a
=q


L
0
c
L
a
ccos
i
dx (61)
h
w
i
q
=
1
2
r

V


L
0
c
L
a
cx
ac
cos
i
dx (62)
h
q
i
a
=q


L
0
c
L
a
eccos
i
dx (63)
h
q
i
q
=
1
2
r

V


L
0
c
L
a
ecx
ac
cos
i
dx (64)
f
w
i
=q


L
0
c
L
0
ccos
2

i
dx

L
0
rgA
i
dx (65)
f
q
i
=q


L
0

c
m
ac
c+c
L
0
e

ccos
2

i
dx+

L
0
rgAe
cg

i
dx (66)
g
w
i
d
k
=q


L
0
c
L
d
k
ccos
2

i
dx (67)
g
q
i
d
k
=q


L
0

c
m
d
k
c+c
L
d
k
e

ccos
2

i
dx (68)
The resultant matrix equation is obtained as
M¨x
e
+C˙x
e
+Kx
e
+Hx
a
=F +Gd (69)
where x
e
=
h
w
1
w
2
   w
n
q
1
q
2
   q
n
i
>
is an elastic state vector of the generalized coordinates,x
a
=
h
a q
i
>
is an aerodynamic state vector of the angle of attack and pitch rate,d =
h
d
1
d
2
   d
n
i
>
is a control
vector of the control surface deflections,M is the generalized mass matrix,C is the generalized damping matrix,K is
the generalized stiffness,H is the generalized aerodynamic coupling matrix,and G is the generalized force derivative
vector due to the flap and slat deflections.
The generalized damping matrix is comprised of both the structural damping and the aerodynamic damping.The
structural damping matrix can be obtained from a modal analysis that transforms the generalized coordinates into the
modal coordinates via the eigenvalue analysis.
Consider the zero-speed structural dynamic equations
¨x
e
+M
1
C
s
˙x
e
+M
1
K
s
x
e
=M
1
F (70)
where C
s
is the structural damping matrix,K
s
is the structural stiffness matrix corresponding to the stiffness matrix K
at zero speed,and F is the force vector.
Assuming that the eigenvalues of the matrix M
1
K
s
are positive real and distinct,then by the similarity transfor-
mation,the matrix M
1
K
s
can be decomposed as
M
1
K
s
=X
2
X
1
(71)
where X is the eigenvector matrix and  =diag(w
1
;w
2
;:::;w
n
) is the diagonal matrix whose elements are the fre-
quencies of the structural modes.
Let q =X
1
x
e
be the modal coordinates,then the transformed structural dynamics equation can be obtained as
¨q+X
1
M
1
C
s
X ˙q+
2
q =X
1
F (72)
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American Institute of Aeronautics and Astronautics
which can be expressed in the modal coordinates as
¨q
i
+2z
i
w
i
˙q
i
+w
2
i
q
i
= f
i
(73)
where z
i
is the damping ratio of the i-th mode.
Let z =diag(z
1
;z
2
;:::;z
n
) be the damping ratio diagonal matrix,then the structural damping matrix is computed
as
C
s
=2MXzX
1
(74)
The total damping matrix includes both the structural damping matrix and the aerodynamic damping matrix ac-
cording to
C =C
s
+C
a
(75)
where C
a
is the aerodynamic damping matrix whose elements are defined by c
q
i
q
j
,c
q
i
w
j
,c
w
i
q
j
,and c
w
i
w
j
.
The aeroelastic modes of the aeroelastic equations are then obtained by the eigenvalue analysis of the following
system:
"
˙x
e
¨x
e
#
=
"
0 I
M
1
K M
1
C
#"
x
e
˙x
e
#
+
"
0
M
1
(Gd Hx
a
)
#
(76)
The flutter boundary is defined to be an airspeed at which the real parts of the eigenvalues of the systems become
zero.
E.Aeroelastic Longitudinal Flight Dynamics
Due to the effect of aeroelasticity,the lift coefficient of an aircraft for symmetric flight can be expressed as
C
L
(t) =C
L
0
+C
L
a
a +C
L
u

V

¯
V

1

+C
L
q
q¯c
2
¯
V

+C
L
d
e
d
e
+
m

k=1
C
L
d
k
d
k
+
n

j=1
C
L
w
j
w
j
(t)
¯c
+
n

j=1
C
L
q
j
q
j
(t) +
n

j=1
C
L
˙w
j
˙w
j
(t)
¯
V

+
n

j=1
C
L
˙
q
j
˙
q
j
(t) ¯c
2
¯
V

(77)
where¯c is the mean aerodynamic chord,
¯
V

is the trimairspeed,andC
L
q
j
,C
L
w
j
,C
L
˙
q
j
,and C
L
˙w
j
are the non-dimensional
aeroelastic lift sensitivities or derivatives which are defined as
C
L
w
j
=
2c
L
a
¯c
S

L
0
tancos
2
c
0
j
dx (78)
C
L
q
j
=
2c
L
a
S

L
0
cos
2
c
j
dx (79)
C
L
˙w
j
=
2c
L
a
¯
V

V

S

L
0
cosc
j
dx (80)
C
L
˙
q
j
=
4c
L
a
¯
V

V

S¯c

L
0
ecosc
j
dx (81)
The drag coefficient due to aeroelasticity may be modeled by a parabolic drag polar
C
D
(t) =C
D
0
+
C
2
L
(t)
pARe
(82)
where AR is the wing aspect ratio,and e is the span efficiency factor.
In addition,the pitching moment coefficient of an aircraft is also influenced by the aeroelastic effects due to
changes in wing lift characteristics.The pitching moment coefficient can be expressed as
C
m
(t) =C
m
0
+C
m
a
a +C
m
u

V

¯
V

1

+C
m
q
q¯c
2
¯
V

+C
m
de
d
e
+
m

k=1
C
m
d
k
d
k
+
n

j=1
C
m
w
j
w
j
(t)
¯c
+
n

j=1
C
m
q
j
q
j
(t) +
n

j=1
C
m
˙w
j
˙w
j
(t)
¯
V

+
n

j=1
C
m
˙
q
j
˙
q
j
(t) ¯c
2
¯
V

(83)
11 of 22
American Institute of Aeronautics and Astronautics
where C
m
q
j
,C
m
w
j
,C
m
˙
q
j
,and C
m
˙w
j
are the non-dimensional aeroelastic pitch moment sensitivities or derivatives
C
m
w
j
=
2c
L
a
S

L
0
x
ac
tancos
2
c
0
j
dx (84)
C
m
q
j
=
2c
L
a
S¯c

L
0
x
ac
cos
2
c
j
dx (85)
C
m
˙w
j
=
2c
L
a
¯
V

V

S¯c

L
0
x
ac
cosc
j
dx (86)
C
m
˙
q
j
=
4c
L
a
¯
V

V

S¯c
2

L
0
x
ac
ecosc
j
dx (87)
The aircraft longitudinal dynamics in the stability axes with b =0,f =0,p =0,and r =0 are then described by
m
˙
V

=C
D
q

S+T cosa mgsin(q a) (88)
mV

˙
a =C
L
q

ST sina +mgcos(q a) (89)
I
YY
˙q =C
m
q

S¯c+
Tz
e
q

S¯c
(90)
˙
q =q (91)
where q is the pitch attitude,S is the aircraft reference wing area,I
YY
is the aircraft moment of inertia about the pitch
axis,T is the thrust force,and z
e
is the offset of the thrust line below the aircraft CG.
III.Aeroelastic Generic Transport Model
Consider the full-scale GTM
8
at a mid-point cruise condition of Mach 0.8 and 30,000 ft with 50%fuel remaining
as shown in Figure 5.
Figure 5 - Generic Transport Model
It is of interest to examine the effect of aeroelasticity on the short period mode of the aircraft.For simplicity,only
the first bending mode (1B) and first torsion mode(1T) are considered.The coupled aeroelastic flight dynamic model
of the GTMcan be expressed in the following state space form:
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American Institute of Aeronautics and Astronautics
2
6
6
6
6
6
6
6
6
4
˙a
˙q
˙w
1
˙
q
1
¨w
1
¨
q
1
3
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
4

1
m
aa
C
L
a
1
m
aa

mV

q

S

C
L
q
¯c
2V



1
m
aa
C
L
w
1
¯c
1
m
qq

C
m
a
+
m
qa
m
aa
C
L
a

1
m
qq
h
C
mq
¯c
2V

+
m
qa
m
aa

C
Lq
¯c
2V


mV

q

S
i
1
m
qq

C
m
w
1
¯c
+
m
qa
m
aa
C
L
w
1
¯c

0 0 0
0 0 0

1
m
w
1
w
1
h
w
1
a

1
m
w
1
w
1
h
w
1
q

1
m
w
1
w
1
k
w
1
w
1

1
m
q
1
q
1
h
q
1
a

1
m
q
1
q
1
h
q
1
q

1
m
q
1
q
1
k
q
1
w
1
+

1
m
aa
C
L
q
1

1
m
aa
C
L
˙w
1
¯
V


1
m
aa
C
L
˙
q
1
¯c
2
¯
V

1
m
qq

C
m
q
1
+
m
qa
m
aa
C
L
q
1

1
m
qq

C
m
˙w
1
¯
V

+
m
qa
m
aa
C
L
˙w
1
¯
V


1
m
qq

C
m
˙
q
1
¯c
2
¯
V

+
m
qa
m
aa
C
L
˙
q
1
¯c
2
¯
V


0 1 0
0 0 1

1
m
w
1
w
1
k
w
1
q
1

1
m
w
1
w
1
c
w
1
w
1

1
m
w
1
w
1
c
w
1
q
1

1
m
q
1
q
1
k
q
1
q
1

1
m
q
1
q
1
c
q
1
w
1

1
m
q
1
q
1
c
q
1
q
1
3
7
7
7
7
7
7
7
7
7
7
7
5
2
6
6
6
6
6
6
6
6
4
a
q
w
q
˙w
˙
q
3
7
7
7
7
7
7
7
7
5
+
2
6
6
6
6
6
6
6
6
6
4

1
m
aa
C
L
d
e

1
m
aa
C
L
d
1
m
qq

C
m
d
e
+
m
qa
m
aa
C
L
d
e

1
m
qq

C
m
d
+
m
qa
m
aa
C
L
d

0 0
0 0
0
1
m
w
1
w
1
g
w
1
d
1
0 
1
m
w
1
w
1
g
q
1
d
1
3
7
7
7
7
7
7
7
7
7
5
"
d
e
d
1
#
(92)
where d
1
is a symmetric control surface on the wing and m
aa
,m
qa
,and m
qq
are defined as
m
aa
=
m
¯
V

q

S
+
C
L
˙a
¯c
2
¯
V

(93)
m
qa
=
C
m
˙a
¯c
2
¯
V

(94)
m
qq
=
I
YY
q

S¯c
(95)
For the configuration with 50% fuel remaining and assuming a structural damping of z
1
= 0:1,the A matrix is
given by
A=
2
6
6
6
6
6
6
6
6
4
8:013410
1
9:657410
1
1:260810
2
5:096610
1
5:463410
4
2:424910
3
2:452610
0
9:146810
1
4:602010
2
2:172610
0
3:516510
3
6:222210
2
0 0 0 0 1 0
0 0 0 0 0 1
1:428510
3
1:586910
1
3:160210
1
1:402910
3
2:436010
0
5:208810
0
3:928210
2
1:892310
0
5:693110
0
2:802810
2
3:227110
1
6:148410
0
3
7
7
7
7
7
7
7
7
5
The eigenvalues of the rigid aircraft’s short period mode can be computed from the 2 by 2 upper left matrix
partition.These eigenvalues are stable
l
SP
=0:85801:5380i
The eigenvalues of the 4 by 4 lower right matrix partition are for the 1B and 1T modes which are also stable
l
1B
=2:09558:2006i
l
1T
=2:196715:1755i
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American Institute of Aeronautics and Astronautics
The eigenvalues of the aeroelastic aircraft are also stable,but with reduced damping in the 1T mode,as seen below
l
SP
=0:50770:5229i
l
1B
=3:18788:3789i
l
1T
=1:454715:1728i
The computed frequencies and damping ratios of the short period mode,and the 1B and 1T modes for the GTM
with 50%fuel remaining are shown in Table 1.
Mode
Short Period
1B
1T
Uncoupled Frequency,rad/sec
1.761
8.4641
15.3337
Coupled Frequency,rad/sec
0.7288
8.9648
15.2424
Uncoupled Damping Ratio
0.4872
0.2476
0.1433
Coupled Damping Ratio
0.6966
0.3556
0.0954
Table 1 - Aeroelastic GTMFrequencies and Damping Ratios at Mach 0.8 and 30,000 ft
The frequencies and damping ratios as a function of the airspeed at the same altitude of 30,000 ft are plotted
in Figures 6 and 7.Generally,the frequencies of the short period mode and 1B mode increase with increasing the
airspeed,while the frequency of the 1T mode decreases precipitously with increasing the airspeed.The divergence
speed is the airspeed at which the torsion modal frequency becomes zero.The damping ratios for both the short period
mode and 1B mode generally increase with increasing the airspeed.The damping ratio for the 1T mode increases with
increasing the airspeed up to Mach 0.7,and thereafter decreases rapidly.The flutter speed is the airspeed at which
the damping ratio of any modes becomes zero.It is apparent that the 1T mode would exhibit a zero damping at a
flutter speed of about Mach 0.85.The lowdamping ratio of the 1T mode can be a problemfor aircraft stability.Active
feedback control can potentially help improve the stability margin of the aeroelastic modes.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
5
10
15
20
25
M

, rad/sec


Short Period Mode
1B Mode
1T Mode
Figure 6 - Frequencies of Aircraft Modes
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American Institute of Aeronautics and Astronautics
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
M




Short Period Mode
1B Mode
1T Mode
Figure 7 - Damping Ratios of Aircraft Modes
IV.Adaptive Control Application
Consider a linearized model of a flexible aircraft with matched uncertainty
˙x =Ax+B
h
u
>
(x
r
)
i
(96)
x
r
=Cx (97)
wherex(t):[0;)!R
n
r
is a state vector that is composed of a rigid-body state vector x
r
(t):[0;)!R
n
r
and a
flexible-body state vector x
e
(t):[0;)!R
n
e
=nn
r
,u(t):[0;)!R
p
is a control vector,A 2R
nn
and B 2R
np
are
constant and known matrices,and 

2R
mp
is a constant and unknown matrix that represents a matched parametric
uncertainty in the rigid-body state,and (x
r
):R
n
r
!R
m
is a vector of known regressors.
The rigid-body dynamics with approximately zero-order flexible dynamics can be obtained by setting ˙x
e
=e (x)
where e is a small parameter
7
"
˙x
r
e
#
=
"
A
rr
A
re
A
er
A
ee
#"
x
r
x
e
#
+
"
B
r
B
e
#
h
u
>
(x
r
)
i
(98)
which yields
x
e
=A
1
ee
e (x) A
1
ee
A
er
x
r
A
1
ee
B
e
h
u
>
(x
r
)
i
Solving for x
e
and substituting it into the rigid-body dynamics yields
˙x
r
=A
p
x
r
+B
p
h
u
>
(x
r
)
i
(x) (99)
where
A
p
=A
rr
A
re
A
1
ee
A
er
(100)
B
p
=B
r
A
re
A
1
ee
B
e
(101)
(x) =A
re
A
1
ee
e (x) (102)
The term (x) represents the effect of unmodeled flexible-body dynamics.The reduced-order plant matrix A
p
is
assumed to be Hurwitz.
The objective is to design an output-feedback adaptive control that enables the rigid-body state vector x
r
(t) to
tracks a reference model
˙x
m
=A
m
x
m
+B
m
r (103)
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American Institute of Aeronautics and Astronautics
where A
m
2R
n
r
n
r
is a known Hurwitz matrix and B
m
2R
n
r
r
is a known.matrix.
The controller is designed with
u =K
x
x
r
+K
r
r 
>
(x
r
) (104)
where (t) is an estimate of 

and it is assumed that K
x
and K
r
can be found such that the following model matching
conditions are satisfied
A
p
B
p
K
x
=A
m
(105)
B
p
K
r
=B
m
(106)
Defined the tracking error as e(t) =x
m
(t) x
r
(t),then the tracking error equation becomes
˙e =A
m
e+B
˜

>
(x
r
) +(x) (107)
where
˜
=

is the estimation error.
Because of the presence of unmodeled dynamics,the standard model-reference adaptive law that adjusts (t)
which is given by
˙
=(x
r
)e
>
PB (108)
is not robust.As the adaptive gain  increases,the adaptive law becomes increasingly sensitive to the unmodeled
dynamics d (x) that can lead to instability.
9
To improve robustness to unmodeled dynamics,we use the optimal control modification adaptive law as proposed
by Nguyen
1
to estimate the unknown parameter 

.The optimal control modification adaptive law
1
is given by
˙
=
h
(x
r
)e
>
PBn(x
r
)
>
(x
r
)B
>
PA
1
m
B
i
(109)
where =
>
>0 2R
mm
is the adaptive gain,n >0 2Ris a tuning parameter,and P is the solution of the Lyapunov
equation
PA
m
+A
>
m
P =Q (110)
As an alternative,the adaptive loop recovery adaptive law as proposed by Calise
2
can be used to adjust (t) as
follows:
˙
=

(x
r
)e
>
PB+h
d(x
r
)
dx
r
d
>
(x
r
)
dx
r


(111)
where h >0 2R is a tuning parameter.
Consider the aeroelastic GTMin the previous section,the reduced-order model of the rigid-body aircraft is given
by
"
˙
a
˙q
#
=
"
0:2187 0:9720
0:4052 0:8913
#"
a
q
#
+
"
0:0651
3:5277
#
d
e
+
h
q

a
q

q
i
"
a
q
#!
+
"

a

a;q;w;q;˙w;
˙
q


q

a;q;w;q;˙w;
˙
q

#
+
"
f
a
(t)
f
q
(t)
#
where q

=0:4 and q

q
=0:2527 represent a parametric uncertainty equivalent to an 100% reduction in the pitch
damping coefficient C
m
q
,and f
a
(t) and f
q
(t) are disturbances due to a moderate vertical wind gust modeled by the
Dryden’s turbulence model
10
with a vertical velocity amplitude of about 10 ft/sec and a pitch rate amplitude of 1.5
deg/sec as shown in Figure 8.
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American Institute of Aeronautics and Astronautics
0
20
40
60
80
100
-1
-0.5
0
0.5
1
t, sec
wa/V
, deg
0
20
40
60
80
100
-2
-1
0
1
2
qa, deg/sec
t, sec
Figure 8 - Vertical Wind Gust Model
A desired reference model of the pitch attitude is given by
¨
q
m
+2zw
n
˙
q
m
+w
2
n
q
m
=w
2
n
r (112)
where z =0:85 and w
n
=1:5 rad/sec are chosen to give a desired handling characteristic.
Let x
r
=
h
a q q
i
>
,u =d
e
,and 
>
=
h
q

a
0 q

q
i
.Anominal controller is designed as u
nom
=K
x
x+
k
r
r where K
x
=
1
b
3
h
a
31
w
2
n
2zw
n
+a
33
i
=
h
0:1149 0:6378 0:4702
i
and k
r
=
1
b
3
w
2
n
=0:6378.The
closed-loop eigenvalues are 0:2112 and 1:27500:7902i.The nominal closed-loop plant is then chosen to be the
reference model as
2
6
4
˙
a
m
˙
q
m
˙q
m
3
7
5
|
{z
}
˙x
m
=
2
6
4
0:2112 0:0415 0:9414
0 0 1
0 2:2500 2:5500
3
7
5
|
{z
}
A
m
2
6
4
a
m
q
m
q
m
3
7
5
|
{z
}
x
m
+
2
6
4
0:0415
0
2:2500
3
7
5
|
{z
}
B
m
r
The optimal control modification and the adaptive loop recovery adaptive laws are blended together in a combined
adaptive law as follows:
˙
=

(x
r
)e
>
PBn(x
r
)
>
(x
r
)B
>
PA
1
m
B+h
d(x
r
)
dx
r
d
>
(x
r
)
dx
r


(113)
where the adaptive gain is chosen to be  = 100I and the input function is chosen as (x
r
) =
h
1 a q q
i
>
whereby the bias input is used to handle the time-varying wind gust disturbances.
For the optimal control modification,the tuning parameters are set to n =0:2 and h =0.For the adaptive loop
recovery,they are set to n =0 and h =0:2.Also the Jacobian of the input function d(x
r
)=dx
r
is simply an identity
matrix,thereby making the adaptive loop recovery effectively a s-modification adaptive law.
11
A pitch attitude doublet is commanded.The response of the aeroelastic GTMwithout adaptive control is plotted
in Figure 9.It is clear that the aircraft response does not track well with the reference model.
17 of 22
American Institute of Aeronautics and Astronautics
0
20
40
60
80
100
-5
0
5
t, sec
, deg




m
0
20
40
60
80
100
-5
0
5
t, sec
, deg




m
0
20
40
60
80
100
-5
0
5
t, sec
q, deg/sec


q
q
m
Figure 9 - Longitudinal Response of Aeroelastic GTMwith No Adaptive Control
Using the standard MRAC by setting n =h =0,the pitch attitude tracking is much improved as shown in Figure
10.However,the pitch rate initial transient is quite large and is characterized with a high frequency signature.In
contrast,with reference to Figure 11,the optimal control modification adaptive law is able to suppress the large initial
transient of the pitch rate and the amplitude of the high frequency content.The response of the aircraft due to the
adaptive loop recovery adaptive law as seen in Figure 12 is very much similar to the optimal control modification
adaptive law.
0
20
40
60
80
100
-5
0
5
t, sec
, deg




m
0
20
40
60
80
100
-5
0
5
t, sec
, deg




m
0
20
40
60
80
100
-5
0
5
t, sec
q, deg/sec


q
q
m
Figure 10 - Longitudinal Response of Aeroelastic GTMwith Standard MRAC ( =100I)
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American Institute of Aeronautics and Astronautics
0
20
40
60
80
100
-5
0
5
t, sec
, deg




m
0
20
40
60
80
100
-5
0
5
t, sec
, deg




m
0
20
40
60
80
100
-5
0
5
t, sec
q, deg/sec


q
q
m
Figure 11 - Longitudinal Response of Aeroelastic GTMwith Optimal Control Modification ( =100I,n =0:2)
0
20
40
60
80
100
-5
0
5
t, sec
, deg




m
0
20
40
60
80
100
-5
0
5
t, sec
, deg




m
0
20
40
60
80
100
-5
0
5
t, sec
q, deg/sec


q
q
m
Figure 12 - Longitudinal Response of Aeroelastic GTMwith Adaptive Loop Recovery ( =100I,h =0:2)
The aeroelastic wing tip bending and torsion deflections are shown in Figures 13 and 14 for four different con-
trollers:baseline nominal control,standard MRAC,optimal control modification adaptive law,and adaptive loop
recovery adaptive law.The aeroelastic GTMis modeled to be rather flexible to demonstrate the aeroelastic effects on
adaptive control.The rigid-body pitch attitude command and wind gust result in a bending deflection amplitude of 5
ft and a torsional deflection amplitude of about 3 deg at the wing tip.The aeroelastic deflections are quite significant
since the flight condition at Mach 0.8 is approaching the flutter speed at Mach 0.85.It is noted that the standard
MRAC results in a very large initial transient of the torsional deflection.This large torsional deflection is clearly not
realistic and in practice would result in excessive wing loading and wing stall.These effects are not taken into account
in the simulations.Nonetheless,this illustrates the undesirable behavior of the standard MRAC in the flight control
implementation for flexible aircraft.
Figure 15 is the plot of the elevator deflections for the four controllers.The standard MRAC produces a significant
control saturation during the initial transient.This saturation leads to undesirable rigid-body aircraft response and
aeroelastic deflections.Both the optimal control modification and adaptive loop recovery adaptive laws produce quite
similar elevator deflections,although it is noted that the deflection is slightly greater in amplitude with the adaptive
loop recovery adaptive law than with the optimal control modification adaptive law.
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American Institute of Aeronautics and Astronautics
0
50
100
-10
-5
0
5
10
t, sec
Wtip, ft


0
50
100
-10
-5
0
5
10
t, sec
Wtip, ft


0
50
100
-10
-5
0
5
10
t, sec
Wtip, ft


0
50
100
-10
-5
0
5
10
t, sec
Wtip, ft


Baseline  =0
MRAC  =100
OCM  =100  =0.2
ALR  =100  =0.2
Figure 13 - Wing Tip Deflection of First Bending Mode
0
50
100
-4
-2
0
2
4
t, sec

tip, deg


0
50
100
-10
-5
0
5
10
t, sec

tip, deg


0
50
100
-4
-2
0
2
4
t, sec

tip, deg


0
50
100
-4
-2
0
2
4
t, sec

tip, deg


MRAC  =100
Baseline  =0
OCM  =100  =0.2
ALR  =100  =0.2
Figure 14 - Wing Tip Twist of First Torsion Mode
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0
50
100
-6
-3
0
3
6
t, sec

e, deg


0
50
100
-20
-10
0
10
20
t, sec

e, deg


0
50
100
-6
-3
0
3
6
t, sec

e, deg


0
50
100
-6
-3
0
3
6
t, sec

e, deg


Baseline  =0
OCM  =100  =0.2
ALR  =100  =0.2
MRAC  =100
Figure 15 - Elevator Deflection
V.Conclusions
This paper presents an aeroelastic model of a generic transport model.Aeroelasticity theory is used to formulate a
coupled bending torsion motion of the aircraft wing as a one-dimensional elastic member.The aeroelastic equations
for the coupled bending torsion motion are solved by a weak-formed formulation using the Galerkin’s method.The
aeroelastic longitudinal dynamic model is then comprised of the longitudinal dynamic model of the rigid-body aircraft
and the aeroelastic model of the flexible-body aircraft wing,both of which are coupled through the angle of attack
and pitch rate.In general,as the airspeed increases,the torsional stiffness decreases,thereby causing the torsional
frequencies to decrease.Moreover,as the airspeed becomes sufficiently fast,the damping ratio of the torsion mode
decreases to zero,at which point a flutter speed is reached.
Adaptive control can be used to accommodate uncertainty for aeroelastic aircraft.An approach based on a reduced-
order model is used to design adaptive controllers.The effect of aeroelasticity is captured in the reduced-order model
as unmodeled dynamics.The standard model-reference adaptive control as well as two recently developed adaptive
laws:optimal control modification and adaptive loop recovery,are implemented.Simulations include a moderate
vertical wing gust Dryden’s model.The results show that the standard MRAC is neither sufficiently robust nor able
to produce well-behaved adaptive signals.Excessive torsional deflections and control saturation due to the standard
MRAC are noted.Both the optimal control modification and adaptive loop recovery adaptive laws are seen to be more
effective in reducing the tracking error while maintaining the aeroelastic deflections to within reasonable levels.
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