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Linear and nonlinear aeroelastic response is examined using a unique test apparatus that allows for experiments of plunge and pitch motion of a wing with prescribed stiffness characteristics. The addition of a control surface, combined with an active control system, extends the stable ight region. Unsteady aerodynamics are modeled with an approximation to Theodorsen's theory appropriate for the low reduced frequencies associated with the experiment. Incorporated with a full-state feedback control law, an optimal observer is utilized to stabilize the system above the open-loop utter velocity. Coulomb damping and hardening of the pitch stiffness are included to examine nonlinear control behavior. The nonlinear model is tested using the control laws developed from an extension of linear theory. Each model is simulated using MATLAB ® and compared with experimental results of the active control system. Excellent correlation between theory and experiment is achieved. Using an optimal observer and full-state feedback, the linear and nonlinear systems are stabilized at velocities that exceed the open-loop utter velocity. Limited control is achieved when the system is undergoing limit cycle oscillations. Nomenclature a = nondimensional distance from the midchord to the elastic axis b = semichord of wing (reference length) C.k/ = Theodorsen's function c = nondimensional distance from midchord to control surface hinge c h = plunge degree of freedom structural damping coef cient c ® = pitch degree of freedom structural damping coef cient e = nondimensional distance from midchord to control surface leading edge g = acceleration due to gravity h = plunge displacement coordinate I ® = mass moment of inertia about the elastic axis K = full-state feedback gains k = reduced frequency .b!=u 1 / k h = plunge degree of freedom structural spring constant k ® = pitch degree of freedom structural spring constant L = estimator gains L.t / = lift of the wing M f = friction moment caused by Coulomb damping M.t / = moment of the wing about the elastic axis m = mass of the wing Q = state weighting matrix N Q = process noise covariance R = control weighting matrix N R = measurement noise covariance u = freestream velocity x ®= nondimensional distance between elastic axis and the center of mass ® = pitch displacement coordinatē = control surface de ection coordinate 1 A = difference in peak amplitudes of the free vibration ¹ h = static coef cient of friction in the plunge direction ¹ ® = static coef cient of friction in the pitch direction ½ = density of air ! = frequency of motion

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The control of nonlinear aeroelastic response of a wing section with a continuous stiffening-type structural nonlinearity is examined through analytical and experimental studies. Motivated by the limited effectiveness of using a single, trailing-edge control surface for the suppression of limit-cycle oscillations of a typical wing section, improvements in the control of limit-cycle oscillations are investigated through the use of multiple control surfaces, namely, an additional leading-edge control surface. The control methodology consists of a feedback linearization approach that transforms the system equations of motion via Lie algebraic methods and a model reference adaptive control strategy that augments the closed-loop system to account for inexact cancellation of nonlinear terms due to modeling uncertainty. Specifically, uncertainty in the nonlinear pitch stiffness is examined. It is shown through simulations and experiments that globally stabilizing control may be achieved by using two control surfaces. Nomenclature a = nondimensional distance from midchord to elastic axis position b = semichord of wing section= pitch cam moment of inertia I cg − wing = wing section moment of inertia about the center of gravity I α = total pitch moment of inertia about elastic axis k h = plunge stiffness m T = total mass of pitch-plunge system m wing = mass of wing section m W − tot = total wing section plus mount mass r cg = distance from elastic axis to center of mass s = wing section span U = freestream velocity x α = nondimensional distance from elastic axis to center of mass α = angle of attack β = trailing-edge control surface deflection γ = leading-edge control surface deflection ρ = air density

The authors examine the stability properties of a class of nonlinear controls derived via feedback linearization techniques for a structurally nonlinear prototypical 2-D wing section. In the case in which the wing section has a single trailing edge control surface, the stability of partial feedback linearization to achieve plunge primary control is studied. It is shown for this case that the zero dynamics associated with the closedloop system response are locally asymptotically stable for a range of ow speeds and elastic axis locations. However, there exist locations of the elastic axis and speeds of the subsonic incompressible ow for which this simple feedback strategy exhibits a wide range of bifurcation phenomena. Both Hopf and pitchfork bifurcations evolve parametrically in terms of the ow speed and elastic axis location. In the case in which the wing section has two control surfaces, the global stability of adaptive control techniques derived from full feedback linearization is studied. In comparison to partial or full feedback linearization techniques, the adaptive control strategies presented do not require explicit knowledge of the form of the structural nonlinearity.