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ANALYSIS AND CONTROL OF NON-AFFINE, NON-STANDARD, SINGULARLY PERTURBED PDF

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ANALYSIS AND CONTROL OF NON-AFFINE, NON-STANDARD, SINGULARLY PERTURBED SYSTEMS A Dissertation by ANSHU NARANG Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Approved by: Chair of Committee, John Valasek Committee Members, Aniruddha Datta Helen L. Reed Srinivas Rao Vadali Department Head, Rodney D. W. Bowersox December 2012 Major Subject: Aerospace Engineering Copyright 2012 Anshu Narang ABSTRACT This dissertation addresses the control problem for the general class of control non-affine, non-standard singularly perturbed continuous-time systems. The prob- lem of control for nonlinear multiple time scale systems is addressed here for the first time in a systematic manner. Toward this end, this dissertation develops the theory of feedback passivation for non-affine systems. This is done by generalizing the Kalman-Yakubovich-Popov lemma for non-affine systems. This generalization is used to identify conditions under which non-affine systems can be rendered pas- sive. Asymptotic stabilization for non-affine systems is guaranteed by using these conditions along with well known passivity-based control methods. Unlike previ- ous non-affine control approaches, the constructive static compensation technique derived here does not make any assumptions regarding the control influence on the nonlinear dynamical model. Along with these control laws, this dissertation presents novel hierarchical control design procedures to address the two major difficulties in control of multiple time scale systems: lack of an explicit small parameter that models the time scale separation and the complexity of constructing the slow man- ifold. These research issues are addressed by using insights from geometric singular perturbation theory and control laws are designed without making any assumptions regarding the construction of the slow manifold. The control schemes synthesized accomplish asymptotic slow state tracking for multiple time scale systems and si- multaneous slow and fast state trajectory tracking for two time scale systems. The control laws are independent of the scalar perturbation parameter and an upper bound for it is determined such that closed-loop system stability is guaranteed. Performance of these methods is validated in simulation for several problems ii from science and engineering including the continuously stirred tank reactor, mag- netic levitation, six degrees-of-freedom F-18/A Hornet model, non-minimum phase helicopter and conventional take-off and landing aircraft models. Results show that the proposed technique applies both to standard and non-standard forms of singu- larlyperturbedsystemsandprovidesasymptotictrackingirrespectiveofthereference trajectory. This dissertation also shows that some benchmark non-minimum phase aerospace control problems can be posed as slow state tracking for multiple time scale systems and techniques developed here provide an alternate method for exact output tracking. iii To dearest Ma, Papa, Amma and Appa and my closest friend and husband, Siddarth iv ACKNOWLEDGEMENTS I am very grateful to my advisor Dr. John Valasek for his constant motivation, support and guidance during my doctorate study. Working with him I have had the opportunity to learn and experience the different aspects of academia that have helped shape my future career aspirations. I am indebted to Dr. Valasek for nomi- nating me as an associate member of the AIAA Guidance, Navigation and Control Technical Committee. Thank you Sir. AspecialthankyoutoDr.SrinivasR.Vadali, Dr.AniruddhaDattaandDr.Helen L. Reed for their critical reviews, feedbacks, and suggestions to improve the quality of my research work. I would like to thank Dr. Suman Chakravorty for sharing his time and expertise through numerous group meetings and classroom discussions, and Dr. James D. Turner for sharing his insights during our discussions in the hallways. I would also like to thank Dr. John E. Hurtado for providing me a firm foundation in dynamical systems theory and giving me the opportunity to work in Land, Air and Space Robotics Laboratory. I am thankful to my undergraduate advisor and mentor, Dr. S. N Saxena for his constant support in all my endeavors and motivating me to perform to the best of my capabilities. My sincere thanks to Ms. Wanda Romero, Ms. Karen Knabe and Ms. Colleen Leatherman for going out of their way to help me throughout my graduate study. I would like to thank the Zonta International Foundation for awarding me the Amelia Earhart Fellowship which was very helpful during my graduate study. I am fortunate to have the friendship and support of Priyanka, Ryan and Neha v who have made life in College Station memorable. I cherish our long chats filled with crazy humor and serious discussions about everything under the sun. It would have been impossible to come this far without the unconditional love and unwavering support of my family. My parents have always been my inspiration and without their motivation I would not have been able to work this far from home. I feel extremely fortunate to have parents whose love has made me what I am today. I am very grateful to Amma and Appa who have supported me and believed in me. Nowords candescribethe influenceSiddarth’sloveand constant wordsofmotivation have had in my life. He has inspired me to excel and stood by me through both good and challenging times. I thank my sister Suruchi, Navin Jijaji, dearest Smridula and Pulkit for their continuous support and love. I feel blessed to be part of such a wonderful family and dedicate this work to them. vi TABLE OF CONTENTS Page ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . v TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Standard and Non-Standard Forms of Singularly Perturbed Systems . 6 1.2 Review of Stabilization Methods for Singularly Perturbed Systems . . 8 1.3 Review of Control Methodologies for Non-Affine in Control Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2. RESEARCH OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 System Class Description . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.1 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Objectives and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Research Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3. A CONSTRUCTIVE STABILIZATION APPROACH FOR OPEN-LOOP UNSTABLE NON-AFFINE SYSTEMS . . . . . . . . . . . . . . . . . . . . 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Passive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Feedback Equivalence to a Passive System/Feedback Passivation . . . 36 3.4 Control Synthesis for Stabilization . . . . . . . . . . . . . . . . . . . . 41 3.4.1 Control Synthesis for Multi-Input Non-Affine Systems . . . . . 41 3.4.2 Construction of Control for Single-Input Non-Affine Systems . 44 vii 3.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5.1 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5.2 One-Dimensional Non-Affine Unstable Dynamics . . . . . . . . 47 3.5.3 Continuously Stirred Tank Reactor . . . . . . . . . . . . . . . 53 3.5.4 Magnetic Levitation System . . . . . . . . . . . . . . . . . . . 59 3.6 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.6.1 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4. ASYMPTOTIC STABILIZATION AND SLOW STATE TRACKING OF CONTROL-AFFINE, TWO TIME SCALE SYSTEMS . . . . . . . . . . . 68 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Approach I: Modified Composite Control . . . . . . . . . . . . . . . . 71 4.3.1 Center Manifold and Its Computation . . . . . . . . . . . . . 71 4.3.2 Control Law Development . . . . . . . . . . . . . . . . . . . . 75 4.3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.4 Approach II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4.1 Control Law Development . . . . . . . . . . . . . . . . . . . . 104 4.4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 109 4.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5. SIMULTANEOUS TRACKING OF SLOW AND FAST TRAJECTORIES FOR CONTROL-AFFINE, TWO TIME SCALE SYSTEMS . . . . . . . . 122 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.2 Control Formulation and Stability Analysis . . . . . . . . . . . . . . . 122 5.2.1 Control Law Development . . . . . . . . . . . . . . . . . . . . 123 5.2.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.3.1 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . 129 5.3.2 Generic Two Degrees-of-Freedom Nonlinear Kinetic Model . . 129 5.3.3 Combined Longitudinal and Lateral/Directional Maneuver for a F/A-18 Hornet . . . . . . . . . . . . . . . . . . . . . . . . . 133 viii 5.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.4.1 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.4.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6. CONTROL OF NONLINEAR, NON-AFFINE, NON-STANDARD MUL- TIPLE TIME SCALE SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . 140 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2 Background: Reduced-Order Models . . . . . . . . . . . . . . . . . . 140 6.3 Control Formulation and Stability Analysis . . . . . . . . . . . . . . . 144 6.3.1 Control Formulation . . . . . . . . . . . . . . . . . . . . . . . 145 6.3.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.4.1 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . 155 6.4.2 Standard Two Time Scale Model . . . . . . . . . . . . . . . . 155 6.4.3 Non-Standard Multiple Time Scale System . . . . . . . . . . . 157 6.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7. SOME APPLICATIONS TO CONTROL OF WEAKLY MINIMUM AND NON-MINIMUM PHASE, NONLINEAR DYNAMICAL SYSTEMS . . . . 163 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.2 The Beam and Ball Experiment . . . . . . . . . . . . . . . . . . . . . 163 7.2.1 Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.2.2 Time Scale Separation Analysis . . . . . . . . . . . . . . . . . 169 7.2.3 Control Formulation . . . . . . . . . . . . . . . . . . . . . . . 171 7.2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 173 7.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.3 HoverControlforanUnmannedThreeDegrees-of-FreedomHelicopter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.3.1 Model Description and Open-Loop Analysis . . . . . . . . . . 181 7.3.2 Time Scale Analysis of the Helicopter Model . . . . . . . . . . 189 7.3.3 Control Formulation and Stability Analysis . . . . . . . . . . . 191 7.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 203 7.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.4 Nap-of-the-Earth Maneuver Control for Conventional Take-off and Landing Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.4.1 Dynamical Model and Open-Loop Analysis . . . . . . . . . . . 212 7.4.2 Time Scale Analysis of the Aircraft Model . . . . . . . . . . . 217 ix 7.4.3 Control Formulation . . . . . . . . . . . . . . . . . . . . . . . 219 7.4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 222 7.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.5.1 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.5.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . 234 8.1 Contributions of Research . . . . . . . . . . . . . . . . . . . . . . . . 234 8.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 8.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 APPENDIX A. REVIEW OF GEOMETRIC SINGULAR PERTURBATION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 APPENDIX B. COMPOSITE LYAPUNOV APPROACH FOR STABILITY ANALYSIS OF SINGULARLY PERTURBED SYSTEMS . . . . . . . . . 262 APPENDIX C. NONLINEAR F/A-18 HORNET AIRCRAFT MODEL . . . 269 x

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ANSHU NARANG. Submitted to the Office of Graduate Studies of. Texas A&M University in partial fulfillment of the requirements for the degree of.
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