Nonlinear Analysis and Control of Aeroelastic Systems Himanshu Shukla Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Mayuresh J. Patil, Chair Craig A. Woolsey Cornel Sultan Michael K. Philen May 3, 2016 Blacksburg, Virginia Keywords: Aeroelasticity, Control, Limit Cycle Oscillations, Optimization, Subcritical, Supercritical Copyright 2016, Himanshu Shukla Nonlinear Analysis and Control of Aeroelastic Systems Himanshu Shukla ABSTRACT Presence of nonlinearities may lead to limit cycle oscillations (LCOs) in aeroelastic systems. LCOs can result in fatigue in wings leading to catastrophic failures. Existence of LCOs for velocities less than the linear flutter velocity has been observed during flight and wind tunnel tests, making such subcritical behavior highly undesirable. The objective of this dissertation is to investigate the existence of subcritical LCOs in aeroelastic systems and develop state feedback controllers to suppress them. The research results are demonstrated on a two degree of freedom airfoil section model with stiffness nonlinearity. Three different approaches are developed and discussed. The first approach uses a feedback linearization controller employing the aeroelastic modal coordinates. The use of modal coor- dinates results in a system which is linearly decoupled making it possible to avoid cancellation of any linear terms when compared to existing feedback linearization controllers which use the physical coordinates. The state and control costs of the developed controller are com- pared to the costs of the traditional feedback linearization controllers. Second approach involves the use of nonlinear normal modes (NNMs) as a tool to predict LCO amplitudes of the aeroelastic system. NNM dynamics along with harmonic balance method are used to generate analytical estimates of LCO amplitude and its sensitivities with respect to the introduced control parameters. A multiobjective optimization problem is solved to gener- ate optimal control parameters which minimize the LCO amplitude and the control cost. The third approach uses a nonlinear state feedback control input obtained as the solution of a multiobjective optimization problem which minimizes the difference between the LCO commencement velocity and the linear flutter velocity. The estimates of LCO commence- ment velocity and its sensitivities are obtained using numerical continuation methods and harmonic balance methods. It is shown that the developed optimal controller eliminates any existing subcritical LCOs by converting them to supercritical LCOs. Nonlinear Analysis and Control of Aeroelastic Systems Himanshu Shukla GENERAL AUDIENCE ABSTRACT Aeroelasticity deals with the problem of fluid-structure interaction in aerial systems. The desire for high speed travel and increased maneuverability has led to the development of complex designs, use of advanced materials and integration of control systems in aerial systems. All these factors cause the system to exhibit diverse response behavior which cannot be captured by linear models in general. The need to analyze and predict the behavior of such systems for design optimization and control design resulted in the use of nonlinear models to represent such systems mathematically. Flutter is a catastrophic phenomenon where the system absorbs energy from the surrounding air and leads to oscillations with increasing amplitudes. Presence of nonlinearities results in limited amplitude flutter or limit cycle oscillations (LCO). The objective of this dissertation is to analyze the effects of stiffness nonlinearities on the response of aeroelastic systems and development of nonlinear controllers to suppress LCOs. A two degree of freedom airfoil section model with quasi- steady aerodynamics is used for this purpose. Three different approaches are discussed and developed. The first approach focuses on development of a feedback linearization controller highlighting the advantages of using modal coordinates in the control design process. The developed controller is optimal for small disturbances and guarantees stability of response for large disturbances. The second and third approaches develop analysis tools to capture response behavior of aeroelastic systems. Using the developed framework analytical estimates of LCO amplitude, LCO commencement velocity and their sensitivities to control parameters are derived. These are used to solve an optimization problem to generate strictly nonlinear feedback control laws to control the LCO amplitudes and eliminate the subcritical LCOs. Attribution Mayuresh J. Patil is co-author for the manuscripts of Chapters 3, 4 and 5. His contributions include strategic advice and review of the articles for technical accuracy, completeness and grammatical correctness. iv Dedicated to my parents, sister and brother Acknowledgements First and foremost, I would like to express my appreciation and gratitude to my advisor, Dr. Mayuresh Patil for all the support and guidance and being a great mentor for me. I want to thank you for encouraging me with research and helping me improve as a researcher. Thanks for guiding me in right direction and meeting with me at even short notices to answer my queries patiently however simple they were. I would also like to thank Dr. Craig Woolsey, Dr. Cornel Sultan and Dr. Michael Philen for serving on my advisory committee. The discussions, comments and brilliant ideas have helped me a lot during the period of my PhD work. I am very thankful to Dr. Mazen Farhood for teaching me a lot about control design. I extend my gratitude to my friends and members of the Aeroelasticity and Flight Dynamics group, Mandar, Eddie, Ryan, Rob, Ben, Patrick, Casey, Chris and Aaron for all the group meetings which taught me a lot about structures. I also want to thank my friends from Nonlinear Systems Lab, Dave, Kyle, Ony, Artur and Ankit for the discussions in Femoyer and NSL and some memorable time. My stay at Blacksburg wouldn’t have been possible without my friends at Blacksburg and I would like to thank Ashok, Gaurav, Riddhika, Sreyoshi, Tamoghna, Vireshwar, Ishan and Dhiraj for all their support and memorable moments during this time. I also want to thank my friends from IIT Kharagpur, Akash, Tanmay, Vimal, Priyesh and Shashank for being just a phone call away and for the few unforgettable trips during this time. vi Finally, I would like to thank my parents, Smt. Geeta Shukla and Shri Ramesh Chandra Shukla, my sister Neetu and my brother Sudhanshu for their love, support and care. This thesis wouldn’t have been possible without your faith in me. vii Contents Contents viii List of Figures xii List of Tables xvii 1 Introduction 1 1.1 AimsandObjectives ............................... 2 1.2 Outline....................................... 4 Bibliography ...................................... 6 2 Literature Review 7 2.1 Aeroelasticity ................................... 7 2.2 Effects of nonlinearities on aeroelastic systems . .... 9 2.3 Predicting response characteristics of aeroelastic systems . ......... 13 2.3.1 Harmonic balance method . 14 2.3.2 Nonlinear normal modes . 15 viii 2.4 Activecontrolofaeroelasticsystems . ... 16 Bibliography ...................................... 23 3 Feedback Linearization based Control of Aeroelastic System represented in Modal Coordinates 32 3.1 Abstract...................................... 32 3.2 Introduction.................................... 33 3.3 Equationsofmotion ............................... 36 3.4 Controlsynthesisformulation . 41 3.4.1 Nonlinear aeroelastic system . 42 3.4.2 Modal transformation to obtain linearly decoupled second order system 42 3.4.3 Feedback Linearization . 46 3.4.4 Stability of the zero dynamics . 50 3.5 Results....................................... 52 3.5.1 Open loop system behavior . 53 3.5.2 Closed loop response - Linear control . 53 3.5.3 Comparison of controller performance . 57 3.6 Conclusion..................................... 63 Bibliography ...................................... 64 4 Controlling Limit Cycle Oscillations Amplitudes in Nonlinear Aeroelastic Systems 68 4.1 Abstract...................................... 68 ix 4.2 Introduction.................................... 69 4.3 Nonlinear Aeroelastic System . 71 4.4 SolutionApproach ................................ 73 4.5 NonlinearNormalModes............................. 75 4.6 Estimation of LCO amplitudes . 79 4.6.1 Harmonic Balance method . 80 4.6.2 Fundamental Harmonic balance for LCO Nonlinear normal mode . 81 4.7 Closedloopsystem ................................ 82 4.7.1 Sensitivity of LCO amplitude to control parameters . 84 4.8 OptimizationFramework............................. 86 4.8.1 Approximatecontrolcost. 86 4.8.2 Optimization Problem . 88 4.9 Results....................................... 89 4.9.1 NNMresponse .............................. 90 4.9.2 Nonlinear Aeroelastic system response . 91 4.10Conclusions .................................... 94 Bibliography ...................................... 96 5 Nonlinear Control Design to Eliminate Subcritical LCOs in Aeroelastic Systems 100 5.1 Abstract...................................... 100 5.2 Introduction.................................... 101 x 5.3 SystemModel................................... 103 5.3.1 Linear Analysis . 105 5.3.2 OpenLoopbehavior ........................... 107 5.4 ProblemStatement ................................ 108 5.4.1 Solution