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 weakformsolution of the aeroelastic equations of the aircraft.The weakformaeroelastic
equations are then coupled with the longitudinal dynamic equations of the rigidbody aircraft to formulate
an aeroelastic ﬂight dynamic model.This model is then used to create a reducedorder state space model of
the rigidbody longitudinal dynamics with the ﬂexiblebody 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 lightweight 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 loadcarrying 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 tradeoff 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 onedimensional structural dynamic theory that models a wing structure as a onedimensional elastic member
in a combined coupled bendingtorsion motion.Aeroelastic analysis is performed based on the quasisteady 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 rigidbody 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 nonrotating earth.The aircraft bodyﬁxed
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
sincos cos sinsin
coscos sin cossin
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 pretwist angle g (x) commonly
designed in many aircraft.Typically,the wing pretwist angle varies frombeing noseup at the wing root to nosedown
at the wing tip.The nosedown pretwist at the wing tip is designed to delay stall onsets.This is called a washout
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 onedimensional elastic approach
can be used to analyze aeroelastic deﬂections with good accuracy.The equivalent onedimensional elastic approach is
a typical formulation in many aeroelasticity studies.
3
It is assumed that the effect of wing curvature is ignored and the
onedimensional 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 pretwist angle,positive nosedown.
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 xaxis,positive nosedown,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
xyV
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
1yV
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 pretwist 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 bendingtorsion 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 pretwisted wing the bending deﬂections W and V are coupled to the torsional
deﬂection via the slope of the wing pretwist 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 deﬂectioninduced 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
wqx
a
+z
1;t
3
7
5
=
2
6
4
usinzW
xt
ucosz
t
wqx
a
+W
t
+y
t
3
7
5
(16)
where u V
,wV
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
=wqx
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
=
wqx
a
+W
t
+y
t
v
x
W
x
v
y
(+g)
ucos
(wqx
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 quarterpoint.On the other
hand,the unsteady aerodynamic method assumes that the unsteady circulation acts at the 3=4chord point.
4
Both
the Theodorsen’s method for simple harmonic airfoil motion
6
and Peters’ ﬁnitestate 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 halfchord length and 1 a 1 is a parameter such that the elastic axis is located at a distance ab
fromthe midchord and a <0 when the elastic axis is forward of the midchord.
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 kth ﬂap,and c
L
d
k
and c
m
d
k
are the section lift and pitching moment control derivative at
the quarterchord point due to the kth 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
rgArAW
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
rgArAW
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 ﬁxedend symmetricmode 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 xcoordinate 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)
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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 jth 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 jth 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 weakformintegral 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
tanq
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
tanq
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 weakformdynamic 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
tanq
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
tanq
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
cosc
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
ctancos
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)
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American Institute of Aeronautics and Astronautics
k
q
i
w
j
=
L
0
EB
2
g
0
0
i
00
j
dxq
L
0
c
L
a
ectancos
2
i
0
j
dx (59)
k
q
i
q
j
=
L
0
GJ +EB
1
g
0
2
0
i
0
j
dxq
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 zerospeed 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 ith 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
=2MXzX
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 nondimensional
aeroelastic lift sensitivities or derivatives which are deﬁned as
C
L
w
j
=
2c
L
a
¯c
S
L
0
tancos
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
cosc
j
dx (80)
C
L
˙
q
j
=
4c
L
a
¯
V
V
S¯c
L
0
ecosc
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)
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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 nondimensional aeroelastic pitch moment sensitivities or derivatives
C
m
w
j
=
2c
L
a
S
L
0
x
ac
tancos
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
cosc
j
dx (86)
C
m
˙
q
j
=
4c
L
a
¯
V
V
S¯c
2
L
0
x
ac
ecosc
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
ST 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 fullscale GTM
8
at a midpoint 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
8:013410
1
9:657410
1
1:260810
2
5:096610
1
5:463410
4
2:424910
3
2:452610
0
9:146810
1
4:602010
2
2:172610
0
3:516510
3
6:222210
2
0 0 0 0 1 0
0 0 0 0 0 1
1:428510
3
1:586910
1
3:160210
1
1:402910
3
2:436010
0
5:208810
0
3:928210
2
1:892310
0
5:693110
0
2:802810
2
3:227110
1
6:148410
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
=0:85801:5380i
The eigenvalues of the 4 by 4 lower right matrix partition are for the 1B and 1T modes which are also stable
l
1B
=2:09558:2006i
l
1T
=2:196715:1755i
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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
=0:50770:5229i
l
1B
=3:18788:3789i
l
1T
=1:454715:1728i
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 rigidbody state vector x
r
(t):[0;)!R
n
r
and a
ﬂexiblebody state vector x
e
(t):[0;)!R
n
e
=nn
r
,u(t):[0;)!R
p
is a control vector,A 2R
nn
and B 2R
np
are
constant and known matrices,and
2R
mp
is a constant and unknown matrix that represents a matched parametric
uncertainty in the rigidbody state,and (x
r
):R
n
r
!R
m
is a vector of known regressors.
The rigidbody dynamics with approximately zeroorder 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 rigidbody 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 ﬂexiblebody dynamics.The reducedorder plant matrix A
p
is
assumed to be Hurwitz.
The objective is to design an outputfeedback adaptive control that enables the rigidbody 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 modelreference 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
>
PBn(x
r
)
>
(x
r
)B
>
PA
1
m
B
i
(109)
where =
>
>0 2R
mm
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 reducedorder model of the rigidbody 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
closedloop eigenvalues are 0:2112 and 1:27500:7902i.The nominal closedloop 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
>
PBn(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 timevarying 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 smodiﬁcation 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.
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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 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 rigidbody 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 rigidbody 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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American Institute of Aeronautics and Astronautics
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 onedimensional elastic member.The aeroelastic equations
for the coupled bending torsion motion are solved by a weakformed formulation using the Galerkin’s method.The
aeroelastic longitudinal dynamic model is then comprised of the longitudinal dynamic model of the rigidbody aircraft
and the aeroelastic model of the ﬂexiblebody 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 reducedorder model
as unmodeled dynamics.The standard modelreference 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 wellbehaved 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.
References
1
Nguyen,N.,Krishnakumar,K.,and Boskovic,J.,“An Optimal Control Modiﬁcation to ModelReference Adaptive Control for Fast
Adaptation,” AIAA Guidance,Navigation,and Control Conference,AIAA 20087283,August 2008.
2
Calise,A.J.,Yucelen,T.,and Muse,J.A.,“A Loop Recovery Method for Adaptive Control,” AIAA Guidance,Navigation,and
Control Conference,AIAA 20095967,August 2009.
3
Hodges,D.H.and Pierce,G.A.,Introduction to Structural Dynamics and Aeroelasticity,Cambridge University Press,2002.
4
Nguyen,N.T.,“Integrated Flight Dynamic Modeling of Flexible Aircraft with Inertial ForcePropulsionAeroelastic Coupling,”
AIAA2008194,46th AIAA Aerospace Sciences Meeting,Reno,NV,Jan 2008.
5
Houbolt,J.C.and Brooks,G.W.,“Differential Equations of Motion for Combined Flapwise Bending,Chordwise Bending,and
Torsion of Twisted NonuniformRotor Blades,” NACA Technical Note 3905,February 1957.
6
Theodorsen,T.and Garrick,I.E.,“Mechanism of Flutter  a Theoretical and Experimental Investigation of the Flutter Problem”,
NACA Report 685,1940.
7
Tuzcu,I.,Nguyen,N.,“Aeroelastic Modeling and Adaptive Control of Generic Transport Model,” AIAA Atmospheric Flight Me
chanics,AIAA20107503,August 2010.
8
Jordan,T.L.,Langford,W.M.,Belcastro,C.M.,Foster,J.M.,Shah,G.H.,Howland,G.,and Kidd,R.,“Development of a
Dynamically Scaled Generic Transport Model Testbed for Flight Research Experiments,” AUVSI Unmanned Unlimited,Arlington,VA,2004.
21 of 22
American Institute of Aeronautics and Astronautics
9
Rohrs,C.E.,Valavani,L.,Athans,M.,and Stein,G.,“Robustness of ContinuousTime Adaptive Control Algorithms in the Presence
of Unmodeled Dynamics,” IEEE Transactions on Automatic Control,Vol AC30,No.9,pp.881889,1985.
10
Schmidt,L.V.,Introduction to Aircraft Flight Dynamics,AIAA Education Series,1998.
11
Ioannu,P.A.and Sun,J.,Robust Adaptive Control,PrenticeHall,1996.
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American Institute of Aeronautics and Astronautics
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