Andrew J. Kurdila

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 Benchmark Control Problem for Supercavitating Vehicles and an Initial Investigation of Solutions

At very high speeds, underwater bodies develop cavitation bubbles at the trailing edges of sharp corners or from contours where adverse pressure gradients are sufficient to induce flow separation. Coupled with a properly designed cavitator at the nose of a vehicle, this natural cavitation can be augmented with gas to induce a cavity to cover nearly the entire body of the vehicle. The formation of the cavity results in a significant reduction in drag on the vehicle and these so-called high-speed supercavitating vehicles (HSSVs) naturally operate at speeds in excess of 75 m s-1. The first part of this paper presents a derivation of a benchmark problem for control of HSSVs. The benchmark problem focuses exclusively on the pitch-plane dynamics of the body which currently appear to present the most severe challenges. A vehicle model is parametrized in terms of generic parameters of body radius, body length, and body density relative to the surrounding fluid. The forebody shape is assumed to be a right cylindrical cone and the aft two-thirds is assumed to be cylindrical. This effectively parametrizes the inertia characteristics of the body. Assuming the cavitator is a flat plate, control surface lift curves are specified relative to the cavitator effectiveness. A force model for a planing afterbody is also presented. The resulting model is generally unstable whenever in contact with the cavity and stable otherwise, provided the fin effectiveness is large enough. If it is assumed that a cavity separation sensor is not available or that the entire weight of the body is not to be carried on control surfaces, limit cycle oscillations generally result. The weight of the body inevitably forces the vehicle into contact with the cavity and the unstable mode; the body effectively skips on the cavity wall. The general motion can be characterized by switching between two nominally linear models and an external constant forcing function. Because of the extremely short duration of the cavity contact, direct suppression of the oscillations and stable planing appear to present severe challenges to the actuator designer. These challenges are investigated in the second half of the paper, along with several approaches to the design of active control systems.

Vision-Based State Estimation for Autonomous Micro Air Vehicles

plane. This paper explores both of these robustness issues using results from a micro air vehicle simulation model developed at the NASA Langley Research Center. In particular, a hierarchy of dynamic models, ranging from a random walk model to a high-fidelity nonlinear micro air vehicle model, is employed in the Kalman filter for a simulated micro air vehicle trajectory with varying levels of measurement noise. It is demonstrated that the visionbased measurement updates in the filter are capable of compensating for significant modeling errors and filter initialization errors. As would be expected, superior overall results are achieved using higher-fidelity dynamic modelsintheKalman filter.Theworkpresentedinthispaperrepresentsthe firststeptowardtheultimateobjectiveof incorporating vision-based state estimation into the design of autonomous flight control systems for micro air vehicles operating in urban environments.

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.

An Investigation of Internal Resonance in Aeroelastic Systems

Although the study of internal resonance in mechanical systems has been given significant consideration, minimal attention has been given to internal resonance for systems which consider the presence of aerodynamic forces. Herein, the investigators examine the possible existence of internal resonances, and the related nonlinear pathologies that such responses may have, for an aeroelastic system which possesses nonlinear aerodynamic loads. Evidence of internal resonance is presented for specific classes of aeroelastic systems, and such adverse response indicates nonlinearities may lead to aeroelastic instabilities that are not predicted by traditional (linear) approaches.

Adaptive Hysteresis Model for Model Reference Control with Actuator Hysteresis

When working with active materials which exhibit profound hysteresis, such as shape memory alloys, the “ perfect” mathematicalrepresentationofthehysteresisdoesnotexist.However,wecanrepresentmanyhysteretictrends by means of operator models that vary in their theoretical, physical, and computational complexity, depending on how precisely they modelthehysteresis. In previousstudies by theauthors,generalized Preisach representations of the hysteresis phenomena by use of Krasnosel’ skii and Pokrovskii (KP) operators have been represented in linear parametric form. This parameterized KP model has been successfully implemented with a gradient-adaptive law foron-lineidentie cationand adaptivecompensation when thehysteresisoutputcanbemeasured. Theapplicability of the parameterized KP model is extended to model reference control systems with hysteresis actuators whose output cannot be measured.

Role of Maggi's Equations in Computational Methods for Constrained Multibody Systems

This paper presents a unified theoretical basis for a class of methods that generate the governing equations of constrained dynamical systems by eliminating the constraints. By using Maggis equations in conjunction with a common projective theory from numerical analysis, it is shown that members of the class are precisely characterized by the basis they choose for the null-space of the variational form of the constraints. For each method considered, the specific basis chosen for the null-space of the variational constraints is derived, as well as a dual basis for the orthogonal complement. The latter basis is of particular interest since it is shown that its knowledge theoretically enables one to generalize certain methods of the class to calculate constraint forces and torques. Practical approaches based on orthogonal transformations to effect this strategy are also outlined. In addition, since the theory presented herein stresses a common, fundamental structure to the various methods, it is especially useful as a means of comparing and evaluating individual numerical algorithms. The theory presented makes clear the relationship between certain numerical instabilities that have been noted in some methods that eliminate a priori constraint contributions to the governing equations by selecting an independent subset of unknowns. It is also briefly indicated how this formalism can be extended, in principle, to the wider class of nonlinear nonholonomic constraints.