The modelfree control of aeroelastic vibrations of a nonlinear 2D wingflap system operating in supersonic flight speed regimes is discussed in this paper. A novel continuous robust controller design yields asymptotically stable vibration suppression in both the pitching and plunging degrees of freedom using the flap deflection as a control input. The controller also ensures that all system states remain bounded at all times during closedloop operation. A Lyapunov method is used to obtain the global asymptotic stability result. The unsteady aerodynamic load is considered by resourcing to the nonlinear Piston Theory Aerodynamics (PTA) modified to account for the effect of the flap deflection. Simulation results demonstrate the performance of the robust control strategy in suppressing dynamic aeroelastic instabilities, such as nonlinear flutter and limit cycle oscillations.
Nomenclature

A, G, GdState and input matrices

Ac, BcState and input matrices of transformed system

A*cMatrix of the system zero dynamics

a∞,p∞, ρ∞Sound speed ,the pressure and air density of the undisturbed flow respectively

BNonlinear restoring moment

C0, C1, C, C3Constants used in the bounded neural network composite weight matrix

e, rTracking error and filtered tracking error, respectively

F, GPositive definite diagonal gain matrixes for update laws ofŴandV

g1, g2Auxiliary saturation gains

Kz, Kv, Γ ,Zb, KdController gains

La(t), Ma(t)lifting and aerodynamic moment

p(y, t)Unsteady pressure

VDimensionless flight speed

vz(t)Downwash velocity normal to the airfoil surface

W, VIdeal neural network interconnection weight matrices
Ŵ
,
Estimated neural network interconnection weight matrices

w(t)Transverse deflection

x, uSystem state and input, respectively

Z, ZBIdeal neural networks composite weight matrix and its bound

Estimated neural networks composite weight matrix and mismatch

Auxiliary control input

ξ Dimensionless plunging displacement ξ =h / b

γ Isentropic gas coefficient (γ = 1.4 for dryair)

τ Dimensionless time τ =Ut/ b

λ Aerodynamic factor

η(t)Vector of system states for analysis of zero dynamics

tr{∙ } Trace of a square matrix defined as the sum of the elements on the matrix main diagonal

∙FFrobenius norm defined as AF=
I. Introduction
In recent years, aeroelastic control and flutter suppression of flexible wings have been extensively investigated by numerous researchers. There are two basic problems associated with the aeroelastic instability of lifting surfaces ? the determination of the flutter boundary and of its character,
i.e
., the identification of the presence of a stable or unstable Limit Cycle Oscillation (LCO) in the proximity of the flutter boundary. Classical flutter analysis is based on the linearized aeroelastic equations, while LCO analysis requires a nonlinear approach
[1]
. The goal of the control is to expand the flight envelope above the uncontrolled flutter instability speed without weight penalties and eventually convert the catastrophic nature of flutter, associated with an unstable LCO typical of a subcritical Hopfbifurcation behavior, into benign flutter, which conversely is associated with a stable LCO typical of a supercritical Hopfbifurcation. A great deal of research activity devoted to the aeroelastic active control and flutter suppression of flight vehicles has been accomplished,
e.g
., see
[2]
. The model nonlinearities can help to stabilize the LCO or be detrimental by destabilizing the LCO
[3]
. The nonlinearities to be included in the aeroelastic model can be structural
[4]
(
i.e
., arising from the kinematic equations); physical
[5]
(
i.e
., those involving the constitutive equations); or aerodynamic appearing in the unsteady aerodynamic equations
[1]
[3]
[6]
. This issue is discussed in the context of panel flutter in
[1]
[6]
[7]
.
A plethora of techniques is available for dealing with the effect of nonlinear structural stiffness in the context of subsonic flow; linear control theory, feedback linearizing techniques, adaptive, and robust control techniques have been employed to account for these nonlinearities,
e.g
.,
[8]

[19]
. Recently, neuralnetworkbased (
i.e
., modelfree) control approaches have been proposed in
[20]
and
[21]
to stabilize a nonlinear aeroelastic wing section. However, there is very little work (
e.g
.,
[3]
,
[22]
) dealing with the aeroelastic vibration suppression for a supersonic wing section in the presence of both structural and aerodynamic nonlinearities.
Motivated by our previous work in
[19]
[21]
and
[23]

[25]
, a novel neural network (NN) based robust controller has been designed to asymptotically stabilize a supersonic aeroelastic system with unstructured nonlinear uncertainties. The nonlinearity of the model depends on the plunging distance and pitching angle. If the nonlinearity is known and could be linearly parameterized, then adaptive control is often considered to be the method of choice. In this paper, we assume unstructured uncertainty in the sense that the structure of the system nonlinearity is considered to be unknown. In contrast to existing neural networkbased controllers that only achieve practical stability, the novel continuous control design in this paper is able to achieve asymptotic stability of the origin. A threelayer neural network is implemented to approximate the unknown nonlinearity of the system. While adaptive control relies on linear parameterizability of the system nonlinearity and the determination of a regression matrix, the universal approximation property of the NN controller enables approximation of the unstructured nonlinear system in a more suitable way. To compensate for the inevitable NN functional approximation error, an integral of a sliding mode term is introduced. Through a Lyapunov analysis, global asymptotic stability can be obtained for the tracking error in the pitching degree of freedom. Then, based on the fact that the system is minimum phase, the asymptotic stability of the plunging degree of freedom is also guaranteed. Simulation results show that this NNbased robust continuous control design can rapidly suppress the flutter and limit cycle oscillations of the aeroelastic system.
The rest of the paper is organized as follows. In Section II, the aeroelastic system dynamics are introduced. In Section III, the control objective is stated explicitly while zero dynamics of the system is analyzed. The openloop error system is developed in Section IV to facilitate the subsequent control design while the closedloop error system is developed in Section V. In Section VI, Lyapunovbased analysis of the stability of the closedloop system is presented while the simulation results are shown in Section VII. Appropriate conclusions are drawn in Section VIII.
II. Model Development
The aeroelastic governing equations of a supersonic wing section with plunging and twisting degreesoffreedom (graphically represented in
Fig. 1
), accounting for flap deflections, and constrained by a linear translational spring and a nonlinear torsional spring, are given as follows
The dimensionless plunging distance (positive downward) is expressed as ξ (≡
h / b
), while
α
is the pitch angle (positive nose up),
are derivatives with respect to dimensionless time τ =
U_{t} / b
, and V =
U
/
bω
_{α}
is the dimensionless flight speed. The parameter
B
represents the nonlinear restoring moment and is defined as the ratio between the linear and nonlinear stiffness coefficients, thus it measures of the degree of nonlinearity of the system;
B
> 0 corresponds to hard structural nonlinearities,
B
= 0 corresponds to a linear model, while
B
< 0 corresponds to soft structural nonlinearities. In addition,
l_{α }
and
m_{α }
represent the dimensionless aerodynamic lift and moment with respect to the elastic axis.
In order to account for flap deflections, some modifications need to be made to the nonlinear Piston Theory Aerodynamics (PTA) which is used here to produce the aerodynamic loads on the lifting surface. To keep the paper selfcontained, a short description of the PTA modified
Supersonic wing section with flap
to account for the flap deflection is presented next. Within the PTA, the unsteady pressure can be defined as follows
where
v_{z}(t)
and
α
_{∞ }
represent the downwash velocity normal to the airfoil surface and the undisturbed speed of sound respectively, and are defined as follows
In the definition of
v_{z}(t)
,
denotes the upper and lower surfaces, respectively, while
U
_{∞ }
denotes the air speed of the undisturbed flow. In the expression (3),
α_{∞}, p_{∞}
and
p_{∞}
denote the pressure and air density of the undisturbed flow, respectively, while
γ
is the isentropic gas coefficient (
γ
= 1.4 for dryair). The transverse deflection
w(t)
in (3) can be expressed as
[26]
where
x
_{0}
and
x
_{1}
denote the dimensionless location of the elastic axis and of the torsional spring of the flap from the leading edge respectively, while
β(t)
represents the flap displacement. In the binomial expansion of (PTA), the pressure formula for PTA in the thirdorder approximation can be obtained by retaining the terms up to and including (
v_{z} / α
_{∞ }
) as follows
[7]
,
[27]

[29]
The aerodynamic correction factor,
is used to correct the PTA to better approximate the pressure at low supersonic flight speed regime. It is important to note that (2) and (5) are only applicable as long as the transformation through contraction and expansion can be consider isentropic,
i.e
., as long as the induced show losses are negligible (lowintensity waves). For more details, see
[1]
[5]
[30]
. PTA provides results in excellent accordance with those based on the Euler solution and the CFL3D code
[31]
. Considering that flow takes place on both the upper and lower surfaces of the airfoil,
U^{+}_{∞}= U^{?}_{∞} = U
; from (3)(5), the aerodynamic pressure
difference can be expressed as
Notice that
δ_{p}
also accounts for the deflection of the flap β. Here,
M = U_{∞} / α_{∞}
is the undisturbed flight Mach number, while
q_{∞} = ρ _{∞}U^{2}_{∞}
/ 2 is the undisturbed dynamic pressure as presented in
[1]
and
[3]
. The model can be simplified to account only for the nonlinearities associated with
α
and discarding those associated with
β
. Even though this is an approximation, the magnitude of the nonlinearities associated with
β
is much smaller than those associated with
α
and will thus be omitted in this paper. In addition, it is assumed in the following development that the nonlinear aerodynamic damping in (6),
i.e
., the terms
w^{3}_{t}, w^{2}_{t} w_{x},
and
w_{t} w^{2}_{t}
will be discarded and consequently, the cubic nonlinear aerodynamic term reduces to
w^{3}_{t}
only. Although nonlinear damping can be included in the model, this paper only considers linear damping and thus conservative estimates of the flutter speed are expected.
Finally, the nonlinear aerodynamic lifting and moment can be obtained from the integration of the difference of pressure on the upper and lower surfaces of the airfoil
where δp+
_{x< bx1}
and δp+
_{x< bx1}
are the aerodynamic pressure difference on the clean airfoil and on the flap. In the governing EOM presented in (1),
l_{α}
and
m_{α}
denote the counterpart of (7) and (8), which are defined as
Here,
μ
represent a the dimensionless mass ratio defined as
m
/ 4
ρb
^{2}
. Given the definitions above, the governing EOM can be transformed into the following form
where
is a vector of systems states,
β(t)
is a flap deflection control input, while
A, G(z), G_{d}(z),
and
Φ
(
y
) are defined as follows
where the explicit definitions for the constants
c_{i}, k_{i}
, ∀
_{i}
= 1, …, 4 as well as
p
_{2}
and
p
_{4}
are reported in the Appendix.
III. Control Objective and Zero Dynamics
The explicit control objective of this paper is to design a modelfree aeroelastic vibration suppression strategy to guarantee the asymptotic convergence of the pitch angle
α
using the flap deflection
β
as a control input. The secondary objective is to ensure that all system states remain bounded at all times during closedloop operation. It is assumed that the measurable variables available for control implementation are the pitch angle
α
, pitch angle velocity
plunging displacement ξ and plunging displacement velocity
Since the proposed control strategy is predicated on the assumption that the system of (12) is minimum phase, the stability of the zero dynamics of the system needs to be assured. For that purpose, the system of (11) is transformed into the following statespace form
Where
=
U^{2}β
is an auxiliary comtrol input,
is a new vector of system states, while
are explicitly defined as follows
where
θ_{i}
∀
_{i}
= 1,2,3,4 are constants that are explicitly defined in the Appendix. In (13) above,
denotes a nonlinearity that encodes the nonlinear structural stiffness. It is to be noted here Φ(0) = 0. The statespace system of (13) can be expanded into the following from
Here, the stability of the zero dynamics is studied for the case when the pitch displacement is
regulated to the origin. Mathematically, this implies that
which implies from the second equation of (15) that
Since Φ
_{1}
(0) = 0. The zero dynamics of the system then reduce to the reduce to the third order system given by
Substituting (16) into the above set of equations for
we obtain the linear system of equations
and A
^{*}
_{c}
is given by
For the nominal system of (15), the eigenvalues of
A^{*}_{c}
lie in the left half plane which implies that the zero dynamics of the system are asymptotically stable,
i.e
., this is a minimum phase system. This implies that asymptotic convergence of the pitching variable α assures the asymptotic convergence of the plunging variable
z
.
IV. OpenLoop Error System Development
Given the definitions of (13) and (14),
can be expressed as follows
The tracking error
is defined where
denotes the desired output vector which needs to be smooth in deference to the requirements of the subsequent control design. For the control objective, one can simply choose
α_{d}
to be zero all the time or use another desirable smooth timevarying trajectory
α_{d}(t)
along which the actual pitching variable
α
can be driven towards the origin. In order to facilitate the ensuing control design and stability analysis, we also define the tracking error
and the filtered tracking error signal
as follows
where
λ
_{1}
,
λ
_{2}
are positive constants. By utilizing the definitions above, one can obtain
By substituting (17) for
in the above expression, the openloop dynamics for
r
can be obtained as follows
After a convenient rearrangement of terms, the openloop dynamics can be rewritten as follows
In order to design a modelfree controller, we define an auxiliary nonlinear signal
N
(·) as follows
By utilizing the definition of (22) above, the openloop dynamics of the system can be compactly
rewritten as follows
V. Control Design and ClosedLoop Error System
Since the structure of the model is assumed to be unknown in the control design, standard adaptive control cannot be applied. In its lieu, a neural network feedforward compensator
along with a robustifying term is proposed to compensate for the function
N
as defined above in (22). By the universal function approximation property
[32]
, the nonlinear function of the system N can be approximated as a threelayer network target function as follows
as long
N
is a general smooth function from
to
and the set of inputs to the function is restricted to a compact set
S
of
. In (24),
denotes the augmented input vector, vector
is the ideal first layer interconnection weight matrix between input layer and hidden layer,
denotes the sigmoidal activation function, while
denotes the ideal second layer interconnection weight matrix. In this work, the weight matrixes
W
and
V
are assumed th be constant and bounded as ∥
W
∥
_{F}
≤
W
_{B}
and ∥
V
∥
_{F}
≤
V
_{B}
, where
W
_{B}
and
V
_{B}
are positive constants. The approximation error is assumed to be bounded in compact set ∥ε∥ < ε
_{N}
where ε
_{N}
is an unknown positive constant related to the number of nodes in the hidden layer.
After substituting the approximation from (24) into (23), one can rewrite the openloop dynamics as follows
where
Motivated by the openloop dynamics and the ensuing stability analysis, the control law is designed as follows
where
K_{v}, K_{d}
> 0 are constant control gains,
is typical threelevel neural network compensator for target function
defined as follows
v
is a robustifying term which will be defined later while
ĝ
is an adaptive estimate for
g
. The dynamic update law for
ĝ
is designed as follows
where the parameter projection operator
proj
{·} is designed to bound
ĝ
in a known compact set Ω such that sgn(
g
_{3}
)
ĝ(t)
≥
ε
> 0 for all time. The projection operator defined here is meaningul because the minimumphase nature of the system ensures that sgn(
g
_{3}
)
g(t)=g
_{3}
^{?1}
g
_{1}
is always positive. In (25),
Ŵ
and
are estimates for the neural network interconnection weight matrices that are dynamically generated as follows
where
and
are postive definite diagonal gain matrixes, while
k
> 0 is a scalar design parameter. By substituting the expression for control law in (26) into the openloop dynamics of (25) and conveniently rearranging the terms, one can obtalin the closedloop system dynamics as follows
where
is a parameter estimation error. Also note that we can write
where the weight estimation errors are defined as
while
w
is defined as follows
To facilitate the subsequent analysis, one can also obtalin a compact form representation for ∥
w
∥ follows
where
C
_{0}
,
C
_{1}
and
C
_{2}
are all positive constants while the ideal composite weight matrix
Z
, estimated composite weight matrix
and the composite weight mismatch matrix
are given as follows
Per the boundedness property for ∥
W
∥
_{F}
and ∥
W
∥
_{F}
as described above, there exists a constant
Z_{B}
such that
Z_{B}
> ∥
Z
∥
_{F}
. Based on the definition
Z_{B}
, the robustifying term
v
can be designed as
Where
K_{z}
is a positive constant. Finally, it is noted that the functional reconstruction error
is assumed to be bounded. Thus, the closedloop dynamics can be finally written as
VI. Stability Analysis
In this section, we provide the stability analysis for the proposed modelfree controller. We begin by defining a nonnegative Lyapunov function candidate
V
_{2}
as follows
After differentiating
V
_{2}
along the closedloop dynamics of r(t) as well as (28), one can obtain the following expression for
After applying the neural network weight update laws designed in (29), canceling out the matched terms and utilizing the definitions of (31), (35) can be upperbounded as
By substituting (30) and (32) into (36), it is possible to further upperbound
as
where the following relation has been used to derive
Based on the fact that
one can choose
K_{z}
>
C
_{2}
such that (37) can be cast as
By defining
C
_{3}
=
Z_{B}
+
C
_{1}
/
k
and conveniently rearranging the terms, (40) yields
By choosing
K_{d}
> [
C_{0} ? kC^{2}_{3}
/ 4], one can obtain the following upperbound on
From (34) and (40), it is easy to see that
r
∈ L
_{2}
∩ L
_{∞ }
while
ĝ
,
The boundedness of
r
implies that
α
,
are bounded by virtue of the definitions of (18) and (19). Since the system is minimum phase and relative degree one, the boundedness of the output guarantees that any first order stable filtering of the input will remain bounded. This implies that all system states remain bounded in closedloop operation which further implies that
stays bounded. Since (26) defines a stable filter acting on a bounded input, it is easy to see that
β
and
stay bounded; furthermore, the flap deflection control input
β
is continuous at all times. The boundedness of
β
implies in turn that
by virtue of the closedloop dynamics of r. Thus, using previous assertions, one can utilize Barbalat's Lemma
[33]
to conclude that
r
→ 0 as
t
→ ∞ which further implies that
From the asymptotic stability of the zero dynamics, we can further guarantee that
x
_{3}
,
x
_{4}
→ 0 as
t
→ ∞ . Thus, both the pitching and plunging variables show asymptotic convergence to the origin.
”. Simulation Result
In this section, simulation results are presented for an aeroelastic system controlled by the proposed continuous robust controller. The nonlinear aerodynamic model is simulated using the dynamics of (1), (7) and (10). The nominal model parameters are list as follows
Controller Parmeters
and the controller parameters are listed in
Table 1
.
The desired trajectory variables
α
_{d}
,
are simply selected as zero. The inital conditions for pitching displacement
α(t)
and plunging displacement
ξ(t)
are chosen as
α
(0) = 5.729deg(about 0.1 radians) and
ξ
(0) = 0 m, while all other state variables are initialized to zero. The initial parameter estimate
ĝ
(0) is set to be 1.20, which is a 10% shift from its nominal value. The flap deflection
β(t)
is constrained to vary between ± 15 deg.
The effect of structural nonlinearities on LCO amplitude was analyzed before applying any control. As shown in
[22]
, increase in structural stiffness factor denoted by B led to decrease in LCO amplitude provided the flutter speed
Openloop dynamics of the aeroelastic system at preflutter speed M=2< M_{flutter}
Closedloop plunging, pitching, control deflection and parameterestimation at preflutter speed, M=2< M_{flutter}
remains constant. Furthermore, we also explored the effect of the location of the elastic axis from the leading edge. It was shown in
[22]
that a decrease in
x
_{0}
leads to decrease in
Openloop dynamics of the aeroelastic system at postflutter speed M=3> M_{flutter}
Closedloop plunging,pitching,control deflections and parameterestimation at postflutter speed, M=3> M_{flutter}
Closedloop plunging, pitching, control deflection and parameterestimation at postflutter speed, M=3> M_{flutter}; control appliedat t=4 s.
LCO amplitude while the flutter speed increases. It was also shown that increasing the damping ratios
ζ_{h}
and
ζ_{α}
resulted in decrease of the amplitude of the LCO.
Fig 2
shows the dynamics of openloop pitching displacement
α
and plunging displacement
ξ
at preflutter speed. The simulation is carried out in subcritical flight speed regime,
M
= 2, below the flutter speed of
M_{flutter}
= 2.15.
Without the controller, it is obvious that the oscillation of pitching degreeoffreedom
α
will converge within 3[s] while the plunging displacement is lightly damped and it takes over 3[s] to converge. In
Fig 3
, it is shown that the proposed robust controller suppresses the oscillation of
α
in less than 1.5[s] while the plunging displacement
ξ
is suppressed in 2.5[s]. The parameter estimate
g
is seen to converge to a constant value within less than 0.5[s].
Another set of simulations is run for postflutter speed. As shown in
Fig 4
, when M is set to be 3, the system dynamics show sustained limit cycle oscillations in openloop operation. Such LCOs is experienced due to the nonlinear pitch stiffness and the aerodynamic nonlinearities. After applying the control to the plant, from
Fig 5
, it is shown that when the control is turned on at t=0[s], the oscillation of
α
is suppressed within 1.5[s]. The dynamic oscillatory behavior of the plunging displacement
ξ
is suppressed within 2.5[s].The control performance is very satisfactory when it start to work at t=0. Next simulation is for delay open of control. In
Fig 6
, control was turned on at t=4[s] after the system had gone into an LCO. It is seen that the oscillations of the pitching degree α and plunging displacement ξ are suppressed respectively in 1.5[s] and 3[s], while in
[13]
convergence time of the two states are 1.5[s] and 4[s]. The parameter estimate
ĝ
also converges to a constant in less than 0.5[s].
These simulation results show that the proposed novel robust controller can effectively suppress the oscillation of both pitching and plunging degreesoffreedom of the airfoil in both preflutter and postflutter flight speed regimes.
VIII. Conclusions
A modular modelfree continuous robust controller was proposed to suppress the aeroelastic vibration characteristics (including flutter and limit cycle oscillations in pre and postflutter
condition) of a supersonic 2DOF lifting surface with flap. Differently from traditional adaptive control strategies, which strictly require the linear parameterization of the system, no prior knowledge of the system model is required for the method presented in this paper. A Lyapunov method based analysis was provided to obtain the global asymptotic stability result. Finally, the simulation results showed that this control strategy can rapidly suppress any aeroelastic vibration
in pre and postflutter flight speed regimes.
 Appendix
The auxiliary constants as well as and that were introduced in the model description of the statespace model are explicitly defined follows
The elements introduced in (15) are explicitly defined as follows
Librescu L.
,
Marzocca P.
,
Silva W.A.
2002
“Supersonic/ Hypersonic Flutter and Postflutter of Geometrically Imperfect Circular Cylindrical Panels”
Journal of Spacecraft and Rockets
39
(5)
802 
812
DOI : 10.2514/2.3882
Mukhopadhyay V.
“Benchmark Active Control Technology”
Journal of Guidance, Control, and Dynamics
Part I 23 (2000) 913960. Part II 23 (2000) 10931139. Part III 24 (2001)
146 
192
DOI : 10.2514/2.4693
Marzocca P.
,
Librescu L.
,
Silva W.A.
2002
“Flutter, Postflutter and Control of a Supersonic 2D Lifting Surface”
Journal of Guidance, Control, and Dynamics
25
(5)
962 
970
DOI : 10.2514/2.4970
Breitbach E.J.
1978
Effects of Structural Nonlinearities on Aircraft Vibration and Flutter
North Atlantic Treaty Organization
Neuilly sur Seine, France
AGARD TR665
Librescu L.
,
Leipholz H.
1975
“Elastostatics and Kinetics of Anisotropic and Heterogeneous ShellType Structures” Aeroelastic Stability of Anisotropic Multilayered Thin Panels
1st ed., Book series
Noordhoff International Publishing
Leyden, The Netherlands
Chapter 1
106158, Appendix A, pp. 543550.
53 
63
Dowell E.H.
,
Ilgamov M.
1988
Studies in nonlinear Aeroelasticity
1st ed
SpringerVerlag
New York
Chapter 2, Chapter 6, pp. 206277.
29 
63
Librescu L.
1965
“Aeroelastic Stability of Orthotropic Heterogeneous Thin Panels in the Vicinity of the Flutter Critical Boundary, Part One: Simply supported panels” Part I
Journal de Mécanique
4
(1)
51 
76
Librescu L
1967
Aeroelastic Stability of Orthotropic Heterogeneous Thin Panels in the Vicinity of the Flutter Critical Boundary, Part Two, Part II
Journal de Mécanique
6
(1)
133 
152
Pak C.
,
Friedmann P.P.
,
Livne E.
1995
“Digital Adaptive Flutter Suppression and Simulation Using Approximate Transonic Aerodynamics”
Journal of Vibration and Control
1
(4)
363 
388
DOI : 10.1177/107754639500100401
Ko J.
,
Strganac T.W.
,
Kurdila A.J.
1999
“Adaptive Feedback Linearization for the Control of a Typical Wing Section with Structural Nonlinearity”
Nonlinear Dynamics
18
289 
301
Xing W.
,
Singh S.N.
2000
“Adaptive Output Feedback Control of a Nonlinear Aeroelastic Structure”
J. Guidance, Control and Dynamics
23
(6)
1109 
1116
DOI : 10.2514/2.4662
Zhang R.
,
Singh S.N.
2001
“Adaptive Output Feedback Control of an Aeroelastic System with Unstructured Uncertainties”
J. Guidance, Control, and Dynamics
3
502 
509
Singh S.N.
,
Wang L.
2002
“Output Feedback form and Adaptive Stabilization of a Nonlinear Aeroelastic System”
J. Guidance, Control and Dynamics
25
(4)
725 
732
DOI : 10.2514/2.4939
Behal A.
,
Marzocca P.
,
Rao V.M.
,
Gnann A.
2006
“Nonlinear adaptive Model Free Control of an Aeroelastic 2D Lifting Surface”
Journal of Guidance, Control and Dynamics
29
(2)
382 
390
DOI : 10.2514/1.14011
Behal A.
,
Rao V. M.
,
Marzocca P.
,
Kamaludeen M.
2006
“Adaptive Control for a Nonlinear Wing Section with Multiple Flaps”
Journal of Guidance, Control, and Dynamics
29
(3)
744 
749
DOI : 10.2514/1.18182
Reddy K.K.
,
Chen J.
,
Behal A.
,
Marzocca P.
2007
“MultiInput/MultiOutput Adaptive Output Feedback Control Design for Aeroelastic Vibration Suppression”
Journal of Guidance, Control, and Dynamics
30
(4)
1040 
1048
DOI : 10.2514/1.27684
Wang Z.
,
Behal A.
,
Marzocca P.
2010
“Adaptive and Robust Aeroelastic Control of Nonlinear Lifting Surfaces with Single/Multiple Control Surfaces: A Review”
International Journal of Aeronautical and Space Science
11
(4)
285 
302
DOI : 10.5139/IJASS.2010.11.4.285
Wang Z.
,
Behal A.
,
Marzocca P.
2011
“Robust Adaptive Output Feedback Control Design for a MIMO Aeroelastic System”
International Journal of Aeronautical and Space Science
12
(2)
157 
167
Zeng J.
,
Wang J.
,
de Callafon R.
,
Brenner M.
2011
“Suppression of the Aeroelastic/Aeroservoelastic Interaction Using Adaptive Feedback Control Instead of Notching Filters”
AIAA Atmospheric Flight Mechanics Conference
Portland, OR
Wang Z.
,
Behal A.
,
Marzocca P.
2012
“Continuous Robust Control for TwoDimensional Airfoils with Leadingand TrailingEdge Flaps”
Journal of Guidance, Control, and Dynamics
35
(2)
510 
519
DOI : 10.2514/1.54347
Zhang F.
,
Söffker D.
2011
“Quadratic Stabilization of a Nonlinear Aeroelastic System Using a Novel NeuralNetworkbased controller”
Science China Technological Sciences
54
(5)
1126 
1133
DOI : 10.1007/s1143101143468
Wang Z.
,
Behal A.
,
Marzocca P.
2011
“ModelFree Control Design for MIMO Aeroelastic System Subject to External Disturbance”
J. of Guidance Control and Dynamics
34
446 
458
DOI : 10.2514/1.51403
Rao V. M.
,
Behal A.
,
Marzocca P.
,
Rubillo C.M.
2006
“Adaptive aeroelastic Vibration Suppression of a Supersonic Airfoil with Flap”
J. Aerospace Science and Technology
10
(4)
309 
315
DOI : 10.1016/j.ast.2006.03.006
Zhang X.
,
Behal A.
,
Dawson D.M.
,
Xian B.
2005
“Output Feedback Control for a Class of Uncertain MIMO Nonlinear Systems With NonSymmetric Input Gain Matrix”
in Proc. of IEEE Conference on Decision and Control
Seville, Spain
7762 
7767
Chen J.
,
Behal A.
,
Dawson D.M.
2008
“Robust Feedback Control for a Class of Uncertain MIMO Nonlinear Systems”
IEEE Transactions on Automatic Control
53
(2)
591 
596
DOI : 10.1109/TAC.2008.916658
Xian B.
,
Dawson D. M.
2004
“de Queiroz, M. S., and Chen, J., A Continuous Asymptotic Tracking Control Strategy for Uncertain Nonlinear Systems”
IEEE Transactions on Automatic Control
49
(7)
1206 
1211
DOI : 10.1109/TAC.2004.831148
Ashley H.
,
Zartarian G.
1956
“Piston Theory  A New Aerodynamic Tool for the Aeroelastician”
Journal of the Aerospace Sciences
23
(10)
1109 
1118
Lighthill M.J.
1953
“Oscillating Airfoils at High Mach Numbers”
Journal of Aeronautical Science
20
(6)
402 
406
Liu D.D.
,
Yao Z.X.
,
Sarhaddi D.
,
Chavez F.R.
1997
“From Piston Theory to a Unified HypersonicSupersonic Lifting Surface Method”
Journal of Aircraft
34
(3)
304 
312
DOI : 10.2514/2.2199
Rodden W.P.
,
Farkas E.F.
,
Malcom H.A.
,
Kliszewski A.M.
1962
Aerodynamic Influence Coefficients from Piston Theory: Analytical Development and Computational Procedure
Aerospace Corporation
TDR169 (323011) TN2
Librescu L.
,
Chiocchia G.
,
Marzocca P.
2003
“Implications of Cubic Physical/Aerodynamic Nonlinearities on the Character of the Flutter Instability Boundary”
International Journal of Nonlinear Mechanics
38
173 
199
DOI : 10.1016/S00207462(01)000543
Thuruthimattam B.J.
,
Friedmann P.P.
,
McNamara J.J.
,
Powell K.G.
2002
“Aeroelasticity of a Generic Hypersonic Vehicle”
43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
April
AIAA Paper no. 20021209
2002 
1209
Krstic M.
,
Kanellakopoulos I.
,
Kokotovic P.
1995
Nonlinear and Adaptive Control Design
John Wiley & Sons
New York