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A Continuous Robust Control Strategy for the Active Aeroelastic Vibration Suppression of Supersonic Lifting Surfaces
A Continuous Robust Control Strategy for the Active Aeroelastic Vibration Suppression of Supersonic Lifting Surfaces
International Journal of Aeronautical and Space Sciences. 2012. Jun, 13(2): 210-220
Copyright ©2012, The Korean Society for Aeronautical Space Science
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/ 3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : March 19, 2012
  • Accepted : April 16, 2012
  • Published : June 30, 2012
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About the Authors
K. Zhang
Department of EECS at the University of Central Florida
kun.zhang@knights.ucf.edu
Z. Wang
Department of EECS at the University of Central Florida
zwang@knights.ucf.edu
A. Behal
Department of EECS and NanoScience Technology Center at the University of Central Florida
abehal@ucf.edu
P. Marzocca
Mechanical and Aeronautical Engineering at Clarkson University
pmarzocc@clarkson.edu

Abstract
The model-free control of aeroelastic vibrations of a non-linear 2-D wing-flap 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 closed-loop operation. A Lyapunov method is used to obtain the global asymptotic stability result. The unsteady aerodynamic load is considered by resourcing to the non-linear 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 non-linear flutter and limit cycle oscillations.
Keywords
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
  • BNon-linear 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 ,
Lager Image
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
Lager Image
  • Estimated neural networks composite weight matrix and mismatch
Lager Image
  • 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
Lager Image
  • 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 ||A||F=
Lager Image
  • Inner product of two matrix, defined astr{B*A}
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 Hopf-bifurcation behavior, into benign flutter, which conversely is associated with a stable LCO typical of a supercritical Hopf-bifurcation. 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 non-linear 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, neural-network-based ( i.e ., model-free) 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 network-based controllers that only achieve practical stability, the novel continuous control design in this paper is able to achieve asymptotic stability of the origin. A three-layer 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 NN-based 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 open-loop error system is developed in Section IV to facilitate the subsequent control design while the closed-loop error system is developed in Section V. In Section VI, Lyapunov-based analysis of the stability of the closed-loop 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 degrees-of-freedom (graphically represented in Fig. 1 ), accounting for flap deflections, and constrained by a linear translational spring and a non-linear torsional spring, are given as follows
Lager Image
The dimensionless plunging distance (positive downward) is expressed as ξ (≡ h / b ), while α is the pitch angle (positive nose up),
Lager Image
are derivatives with respect to dimensionless time τ = Ut / b , and V = U / α is the dimensionless flight speed. The parameter B represents the non-linear restoring moment and is defined as the ratio between the linear and non-linear 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, lα and mα 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 non-linear Piston Theory Aerodynamics (PTA) which is used here to produce the aerodynamic loads on the lifting surface. To keep the paper self-contained, a short description of the PTA modified
Lager Image
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
Lager Image
where vz(t) and α represent the downwash velocity normal to the airfoil surface and the undisturbed speed of sound respectively, and are defined as follows
Lager Image
In the definition of vz(t) ,
Lager Image
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 dry-air). The transverse deflection w(t) in (3) can be expressed as [26]
Lager Image
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 third-order approximation can be obtained by retaining the terms up to and including ( vz / α ) as follows [7] , [27] - [29]
Lager Image
The aerodynamic correction factor,
Lager Image
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 (low-intensity 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
Lager Image
difference can be expressed as
Lager Image
Notice that δp also accounts for the deflection of the flap β. Here, M = U / α is the undisturbed flight Mach number, while q = ρ U2 / 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 w3t, w2t wx, and wt w2t will be discarded and consequently, the cubic nonlinear aerodynamic term reduces to w3t 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
Lager Image
Lager Image
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), lα and mα denote the counterpart of (7) and (8), which are defined as
Lager Image
Lager Image
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
Lager Image
where
Lager Image
is a vector of systems states, β(t) is a flap deflection control input, while A, G(z), Gd(z), and Φ ( y ) are defined as follows
Lager Image
where the explicit definitions for the constants ci, ki , ∀ 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 model-free 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 closed-loop operation. It is assumed that the measurable variables available for control implementation are the pitch angle α , pitch angle velocity
Lager Image
plunging displacement ξ and plunging displacement velocity
Lager Image
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 state-space form
Lager Image
Where
Lager Image
= U2β is an auxiliary comtrol input,
Lager Image
is a new vector of system states, while
Lager Image
are explicitly defined as follows
Lager Image
where θi i = 1,2,3,4 are constants that are explicitly defined in the Appendix. In (13) above,
Lager Image
denotes a nonlinearity that encodes the nonlinear structural stiffness. It is to be noted here Φ(0) = 0. The state-space system of (13) can be expanded into the following from
Lager Image
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
Lager Image
which implies from the second equation of (15) that
Lager Image
Since Φ 1 (0) = 0. The zero dynamics of the system then reduce to the reduce to the third order system given by
Lager Image
Substituting (16) into the above set of equations for
Lager Image
we obtain the linear system of equations
Lager Image
and A * c is given by
Lager Image
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. Open-Loop Error System Development
Given the definitions of (13) and (14),
Lager Image
can be expressed as follows
Lager Image
The tracking error
Lager Image
is defined where
Lager Image
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 αd to be zero all the time or use another desirable smooth time-varying trajectory αd(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
Lager Image
and the filtered tracking error signal
Lager Image
as follows
Lager Image
Lager Image
where λ 1 , λ 2 are positive constants. By utilizing the definitions above, one can obtain
Lager Image
By substituting (17) for
Lager Image
in the above expression, the open-loop dynamics for r can be obtained as follows
Lager Image
After a convenient rearrangement of terms, the open-loop dynamics can be rewritten as follows
Lager Image
In order to design a model-free controller, we define an auxiliary nonlinear signal N (·) as follows
Lager Image
By utilizing the definition of (22) above, the open-loop dynamics of the system can be compactly
rewritten as follows
Lager Image
V. Control Design and Closed-Loop 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
Lager Image
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 three-layer network target function as follows
Lager Image
as long N is a general smooth function from
Lager Image
to
Lager Image
and the set of inputs to the function is restricted to a compact set S of
Lager Image
. In (24),
Lager Image
denotes the augmented input vector, vector
Lager Image
is the ideal first layer interconnection weight matrix between input layer and hidden layer,
Lager Image
denotes the sigmoidal activation function, while
Lager Image
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 open-loop dynamics as follows
Lager Image
where
Lager Image
Motivated by the open-loop dynamics and the ensuing stability analysis, the control law is designed as follows
Lager Image
where Kv, Kd > 0 are constant control gains,
Lager Image
is typical three-level neural network compensator for target function
Lager Image
defined as follows
Lager Image
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
Lager Image
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 minimum-phase nature of the system ensures that sgn( g 3 ) g(t)=g 3 ?1 g 1 is always positive. In (25), Ŵ and
Lager Image
are estimates for the neural network interconnection weight matrices that are dynamically generated as follows
Lager Image
where
Lager Image
and
Lager Image
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 open-loop dynamics of (25) and conveniently rearranging the terms, one can obtalin the closed-loop system dynamics as follows
Lager Image
where
Lager Image
is a parameter estimation error. Also note that we can write
Lager Image
where the weight estimation errors are defined as
Lager Image
while w is defined as follows
Lager Image
To facilitate the subsequent analysis, one can also obtalin a compact form representation for ∥ w ∥ follows
Lager Image
where C 0 , C 1 and C 2 are all positive constants while the ideal composite weight matrix Z , estimated composite weight matrix
Lager Image
and the composite weight mismatch matrix
Lager Image
are given as follows
Lager Image
Per the boundedness property for ∥ W F and ∥ W F as described above, there exists a constant ZB such that ZB > ∥ Z F . Based on the definition ZB , the robustifying term v can be designed as
Lager Image
Where Kz is a positive constant. Finally, it is noted that the functional reconstruction error
Lager Image
is assumed to be bounded. Thus, the closed-loop dynamics can be finally written as
Lager Image
VI. Stability Analysis
In this section, we provide the stability analysis for the proposed model-free controller. We begin by defining a nonnegative Lyapunov function candidate V 2 as follows
Lager Image
After differentiating V 2 along the closed-loop dynamics of r(t) as well as (28), one can obtain the following expression for
Lager Image
Lager Image
Lager Image
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
Lager Image
By substituting (30) and (32) into (36), it is possible to further upperbound
Lager Image
as
Lager Image
where the following relation has been used to derive
Lager Image
Based on the fact that
Lager Image
one can choose
Kz > C 2 such that (37) can be cast as
Lager Image
By defining C 3 = ZB + C 1 / k and conveniently rearranging the terms, (40) yields
Lager Image
By choosing Kd > [ C0 ? kC23 / 4], one can obtain the following upperbound on
Lager Image
Lager Image
From (34) and (40), it is easy to see that r ∈ L 2 ∩ L while ĝ ,
Lager Image
The boundedness of r implies that α ,
Lager Image
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
Lager Image
stays bounded. Since (26) defines a stable filter acting on a bounded input, it is easy to see that β and
Lager Image
stay bounded; furthermore, the flap deflection control input β is continuous at all times. The boundedness of β implies in turn that
Lager Image
by virtue of the closed-loop 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
Lager Image
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
Lager Image
Controller Parmeters
Lager Image
and the controller parameters are listed in Table 1 .
The desired trajectory variables α d ,
Lager Image
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
Lager Image
Open-loop dynamics of the aeroelastic system at pre-flutter speed M=2< Mflutter
Lager Image
Closed-loop plunging, pitching, control deflection and parameterestimation at pre-flutter speed, M=2< Mflutter
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
Lager Image
Open-loop dynamics of the aeroelastic system at post-flutter speed M=3> Mflutter
Lager Image
Closed-loop plunging,pitching,control deflections and parameterestimation at post-flutter speed, M=3> Mflutter
Lager Image
Closed-loop plunging, pitching, control deflection and parameterestimation at post-flutter speed, M=3> Mflutter; control appliedat t=4 s.
LCO amplitude while the flutter speed increases. It was also shown that increasing the damping ratios ζh and ζα resulted in decrease of the amplitude of the LCO.
Fig 2 shows the dynamics of open-loop pitching displacement α and plunging displacement ξ at pre-flutter speed. The simulation is carried out in subcritical flight speed regime, M = 2, below the flutter speed of Mflutter = 2.15.
Without the controller, it is obvious that the oscillation of pitching degree-of-freedom 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 post-flutter speed. As shown in Fig 4 , when M is set to be 3, the system dynamics show sustained limit cycle oscillations in open-loop operation. Such LCOs is experienced due to the non-linear 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 degrees-of-freedom of the airfoil in both pre-flutter and post-flutter flight speed regimes.
VIII. Conclusions
A modular model-free continuous robust controller was proposed to suppress the aeroelastic vibration characteristics (including flutter and limit cycle oscillations in pre- and post-flutter
condition) of a supersonic 2-DOF 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 post-flutter flight speed regimes.
- Appendix
The auxiliary constants as well as and that were introduced in the model description of the state-space model are explicitly defined follows
Lager Image
The elements introduced in (15) are explicitly defined as follows
Lager Image
References
Librescu L. , Marzocca P. , Silva W.A. 2002 “Supersonic/ Hypersonic Flutter and Post-flutter 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) 913-960. Part II 23 (2000) 1093-1139. 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 2-D Lifting Surface” Journal of Guidance, Control, and Dynamics 25 (5) 962 - 970    DOI : 10.2514/2.4970
Breitbach E.J. 1978 Effects of Structural Non-linearities on Aircraft Vibration and Flutter North Atlantic Treaty Organization Neuilly sur Seine, France AGARD TR-665
Librescu L. , Leipholz H. 1975 “Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures” Aeroelastic Stability of Anisotropic Multilayered Thin Panels 1st ed., Book series Noordhoff International Publishing Leyden, The Netherlands Chapter 1 106-158, Appendix A, pp. 543-550. 53 - 63
Dowell E.H. , Ilgamov M. 1988 Studies in non-linear Aeroelasticity 1st ed Springer-Verlag New York Chapter 2, Chapter 6, pp. 206-277. 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 Non-linearity” Non-linear Dynamics 18 289 - 301
Xing W. , Singh S.N. 2000 “Adaptive Output Feedback Control of a Non-linear 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 Non-linear 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 2-D 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 “Multi-Input/Multi-Output 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 Two-Dimensional Airfoils with Leadingand Trailing-Edge 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 Neural-Networkbased controller” Science China Technological Sciences 54 (5) 1126 - 1133    DOI : 10.1007/s11431-011-4346-8
Wang Z. , Behal A. , Marzocca P. 2011 “Model-Free 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 Non-Symmetric 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 Hypersonic-Supersonic 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 TDR-169 (3230-11) TN-2
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/S0020-7462(01)00054-3
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. 2002-1209 2002 - 1209
Hornik K. , Stinchombe M. , White H. 1989 “Multilayer Feedforward Networks are Universal Approximators” J. Neural Networks 2 359 - 366    DOI : 10.1016/0893-6080(89)90020-8
Krstic M. , Kanellakopoulos I. , Kokotovic P. 1995 Nonlinear and Adaptive Control Design John Wiley & Sons New York