Jeonghwan Ko

Nonlinear Control of a Prototypical Wing Section with Torsional Nonlinearity

With the increase in popularity of active materials for control actuation, renewed interest is evident in the derivation of control methodologies for aeroelastic systems. It has been known for some time that prototypical aeroelastic wing sections can exhibit a broad class of pathological response regimes when the system includes certaintypesofnonlinearities.Weinvestigatenonlinearcontrollawsforaeroelasticsystemsthatincludepolynomial structural nonlinearities and study the closed-loop stability of the system. It is shown that locally asymptotically stable(nonlinear)feedbackcontrollerscanbederivedfortheaeroelasticsystemusingpartialfeedbacklinearization techniques. In this case, the stability results are necessarily local in nature and are derived by considering stability of theassociated zero dynamics subsystem. Itis also demonstrated that globally stable (nonlinear)adaptivecontrol methods can be derived for a class of aeroelastic systems under consideration. Numerical simulations are used to provide empirical validation of some of the results.

IDENTIFICATION AND CONTROL OF LIMIT CYCLE OSCILLATIONS IN AEROELASTIC SYSTEMS

Nonlinearities in the aeroelastic system induce pathologies such as the observed store-induced limit cycle oscillations found with certain high-performance aircraft configurations. Many prior studies, including efforts by these authors, focus on the nonlinear behavior of the uncontrolled, nonlinear aeroelastic system. These studies are briefly reviewed. More importantly, there is limited study for the active control of these nonlinear aeroelastic systems. Although a linear controller may stabilize the nonlinear system under some circumstances, empirical evidence suggests that these control methods -will prove unreliable in strongly nonlinear regimes and that stability is not guaranteed. Herein, the authors describe the development of control strategies appropriate for these nonlinear systems. A nonlinear controller, and the resulting closed-loop stability, based on a partial feedback linearization are discussed. The approach depends upon the exact cancellation of the nonlinearity and, as a co_nsequence, the authors introduce an adaptive method in which guarantees of stability are evident. The authors present experimental results obtained using the adaptive controller.

Stability and Control of a Structurally Nonlinear Aeroelastic System

The authors examine the stability properties of a class of nonlinear controls derived via feedback linearization techniques for a structurally nonlinear prototypical two-dimensional 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 closed-loop system response are locally asymptotically stable for a range of e ow speeds and elastic axis locations. However, there exist locations of the elastic axis and speeds of the subsonic/incompressible e 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 e 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 with partial or full feedback linearization techniques, the adaptive control strategies presented do not require explicit knowledge of the form of the structural nonlinearity.

Adaptive Feedback Linearization for the Control of a Typical Wing Section with Structural Nonlinearity

Earlier results by the authors showed constructions of Lie algebraic, partial feedback linearizing control methods for pitch and plunge primary control utilizing a single trailing edge actuator. In addition, a globally stable nonlinear adaptive control method was derived for a structurally nonlinear wing section with both a leading and trailing edge actuator. However, the global stability result described in a previous paper by the authors, while highly desirable, relied on the fact that the leading and trailing edge actuators rendered the system exactly feedback linearizable via Lie algebraic methods. In this paper, the authors derive an adaptive, nonlinear feedback control methodology for a structurally nonlinear typical wing section. The technique is advantageous in that the adaptive control is derived utilizing an explicit parameterization of the structural nonlinearity and a partial feedback linearizing control that is parametrically dependent is defined via Lie algebraic methods. The closed loop stability of the system is guaranteed to be stable via application of La Salles invariance principle.

A class of finite element methods based on orthonormal, compactly supported wavelets

This paper develops a class of finite elements for compactly supported, shift-invariant functions that satisfy a dyadic refinement equation. Commonly referred to as wavelets, these basis functions have been shown to be remarkably well-suited for integral operator compression, but somewhat more difficult to employ for the representation of arbitrary boundary conditions in the solution of partial differential equations. The current paper extends recent results for treating periodized partial differential equations on unbounded domains in Rn, and enables the solution of Neumann and Dirichlet variational boundary value problems on a class of bounded domains. Tensor product, wavelet-based finite elements are constructed. The construction of the wavelet-based finite elements is achieved by employing the solution of an algebraic eigenvalue problem derived from the dyadic refinement equation characterizing the wavelet, from normalization conditions arising from moment equations satisfied by the wavelet, and from dyadic refinement relations satisfied by the elemental domain. The resulting finite elements can be viewed as generalizations of the connection coefficients employed in the wavelet expansion of periodic differential operators. While the construction carried out in this paper considers only the orthonormal wavelet system derived by Daubechies, the technique is equally applicable for the generation of tensor product elements derived from Coifman wavelets, or any other orthonormal compactly supported wavelet system with polynomial reproducing properties.

Nonlinear control theory for a class of structural nonlinearities in a prototypical wing section

With the increase in popularity of active materials for control actuation, renewed interest is evident in the derivation of control methodologies for aeroelastic systems. It has been known for some time that prototypical aeroelastic wing sections can exhibit a broad class of pathological response regimes when the system includes certain types of nonlinearities. In this paper, we investigate nonlinear control laws for aeroelastic systems that include polynomial structural nonlinearities, and study the closed loop stability of the system. It is shown that locally asymptotically stable (nonlinear) feedback controllers can be derived for the aeroelastic system using partial feedback linearization techniques. In this case, the stability results are necessarily local in nature and are derived by considering stability of the associated zero dynamics subsystem. It is also demonstrated that globally stable (nonlinear) adaptive control methods can be derived for a class of aeroelastic systems under consideration. Numerical simulations are used to provide empirical validation of some of the results in this paper.

Reduced Order Nonlinear Navier-Stokes Models for Synthetic Jets

While the potential for the use of synthetic jet actuators to achieve flow control has been noted for some time, most of such flow control studies have been empirical or experimental in nature. Several technical issues must be resolved to achieve rigorous, model-based, closed-loop control methodologies for this class of actuators. First, we seek to derive and evaluate model order reduction methods based on proper orthogonal decomposition that are suitable for synthetic jet actuators. Second, we seek to derive rigorously stable feedback control laws for the derived reduced order models

Nonlinear Dynamics and Control for a Structurally Nonlinear Aeroelastic System

This paper studies the stability properties of a class of nonlinear controls derived via feedback linearization techniques for a structurally nonlinear prototypical 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 that the zero dynamics in this case associated with the closed loop system response is locally asymptotically stable for a range of flow speeds and elastic axis locations. However, there exist locations of the elastic axis and speeds of the flow for which this simple feedback strategy exhibits a wide range of bifurcation phenomena. Both Hopf and pitchfolk bifurcations evolve parametrically in terms of the flow 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.

Structured Model Reference Adaptive Control for a Wing Section with Structural Nonlinearity

The recently derived structured model reference adaptive control (SMRAC) method is considered for the active suppression of limit cycle oscillations for a typical wing section with a structural nonlinearity. The SMRAC method uses the specific structure of the equations of motion governing general mechanical systems. This adaptive control method is implemented and tested in wind tunnel experiments to validate the performance. For comparison, results obtained with adaptive feedback linearization are also presented. It is shown that the derived SMRAC is advantageous in that it suppresses limit cycle oscillations at higher velocities, and also it can rigorously treat actuator saturation a priori.

Particle image velocimetry via wavelet analysis

Novel analytical and image processing methods derived as part of the development of a digital particle image velocimetry system, based on multiresolution analysis, are presented. The derivation of wavelet-based multiresolution methods for velocity e eld reconstruction is addressed, and the experimental setup used for validation of results is described. New techniques are proposed with improved spatial resolution and reliability over some existing methods. The techniques are based on wavelet-based representations of digital particle image data that are used to calculate spatially localized and frequency localized e ltered correlations of successive images. An essential feature of the method is the development of windowed cross-correlation expressions for wavelet-based expansions that are not orthogonal (or biorthogonal) over the cross-correlation window. The methodology makes use of recently introduced ree nable functions and generalized connection coefe cients derived in wavelet-based e nite element methods. A conventional charge-coupled device camera is used with a frame rate of 30 frames/s and pixel resolution of 512 £ 480 per frame. The images are acquired in pairs at 30 frames/s, with a user-dee ned time delay between image pairs, to capture the e owe eld structure evolution. The e ow illumination was achieved using a 5-W, argon-ion laser. Finally, hardware and algorithm performance is demonstrated via sample water-tunnel experiments.