C++ FRAMEWORK FOR SOLVING NONLINEAR OPTIMAL CONTROL PROBLEMS By YUNUS AGAMAWI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2019 c 2019 Yunus Agamawi Dedicated to breaking the barrier. ACKNOWLEDGMENTS I would like to express my sincerest gratitude and appreciation to the many people who have helped me throughout this journey. First and foremost, I would like to thank my advisor, Dr. Anil Rao, for giving me the opportunity to earn a PhD at the University of Florida. His expertise and guidance throughout my graduate career have been invaluable. I would also like to thank the members of my supervisory committee, Dr. William Hager, Dr. Riccardo Bevilacqua, and Dr. John Conklin, for their time and support throughout the doctoral process. I would like to thank my fellow VDOL members, Joseph Eide, Alex Miller, Mitzi Dennis, Rachel Keil, Elisha Pager, and Brittanny Holden, as well, for making the field of computational research ever so slightly more bearable. Specifically, I'd like to thank Joe and Alex for always being willing to critically think with me, and to thank Mitzi, Rachel, Elisha, and Brittanny for creating such a copacetic work environment. Having intellectual peers who could understand and relate to the arduous process of learning and applying optimal control and computational theory was vital to maintaining my sanity. Last, but certainly not least, I'd like to thank my loving family. My parents have been nothing but completely supportive and approving of my academic ambitions since I was in preschool. I thank my mother for teaching me the importance of dedication and hard work, and I thank my father for inspiring me to be the best version of myself that I can be. I would also like to thank my big brother, Yusuf, for being a constant beacon of success with which to compete, and my amazing girlfriend, Amie, for her boundless patience, kindness, and love. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................4 LIST OF TABLES.....................................8 LIST OF FIGURES.................................... 10 ABSTRACT........................................ 12 CHAPTER 1 INTRODUCTION.................................. 14 2 MATHEMATICAL BACKGROUND........................ 25 2.1 Bolza Optimal Control Problem....................... 25 2.2 Numerical Methods for Differential Equations................ 28 2.2.1 Explicit Simulation (Time-Marching)................. 29 2.2.2 Implicit Simulation (Collocation)................... 32 2.3 Numerical Methods for Optimal Control................... 34 2.3.1 Indirect Methods............................ 34 2.3.1.1 Indirect Shooting....................... 35 2.3.1.2 Indirect Multiple-Shooting.................. 36 2.3.1.3 Indirect Collocation..................... 37 2.3.2 Direct Methods............................. 38 2.3.2.1 Direct Shooting........................ 40 2.3.2.2 Multiple Direct Shooting................... 40 2.3.2.3 Direct Collocation...................... 41 2.4 Family of Legendre-Gauss Direct Collocation Methods........... 42 2.4.1 Transformed Continuous Bolza Problem................ 42 2.4.2 LG, LGR, and LGL Collocation Points................ 43 2.4.3 Legendre-Gauss-Lobatto Orthogonal Collocation Method...... 45 2.4.4 Legendre-Gauss Orthogonal Collocation Method........... 46 2.4.5 Legendre-Gauss-Radau Orthogonal Collocation Method....... 49 2.4.6 Benefits of Using Legendre-Gauss-Radau Collocation Method.... 52 2.5 Numerical Optimization............................ 53 2.5.1 Unconstrained Optimization...................... 53 2.5.2 Equality Constrained Optimization.................. 54 2.5.3 Inequality Constrained Optimization................. 56 3 EXPLOITING SPARSITY IN LEGENDRE-GAUSS-RADAU COLLOCATION METHOD....................................... 58 3.1 Notation and Conventions........................... 58 3.2 General Multiple-Phase Optimal Control Problem.............. 61 3.3 Variable-Order Legendre-Gauss-Radau Collocation Method......... 65 5 3.4 Form of NLP Resulting from LGR Transcription............... 71 3.4.1 NLP Decision Vector Arising from LGR Transcription........ 72 3.4.2 NLP Objective Function Arising from LGR Transcription...... 72 3.4.3 Gradient of NLP Objective Function................. 73 3.4.4 NLP Constraint Vector Arising from LGR Transcription....... 74 3.4.5 NLP Constraint Jacobian........................ 74 3.4.6 NLP Lagrangian Arising from LGR Transcription.......... 80 3.4.7 NLP Lagrangian Hessian........................ 82 3.5 Conclusions................................... 83 4 COMPARISON OF DERIVATIVE ESTIMATION METHODS IN SOLVING OPTIMAL CONTROL PROBLEMS USING DIRECT COLLOCATION.... 84 4.1 General Multiple-Phase Optimal Control Problem............. 88 4.2 Legendre-Gauss-Radau Collocation...................... 89 4.3 Nonlinear Programming Problem Arising from LGR Collocation...... 93 4.4 Derivative Approximation Methods...................... 95 4.4.1 Finite-Difference Methods....................... 95 4.4.2 Bicomplex-step Derivative Approximation.............. 96 4.4.3 Hyper-Dual Derivative Approximation................ 99 4.4.4 Automatic Differentiation....................... 101 4.5 Comparison of Various Derivative Approximation Methods......... 102 4.5.1 Derivative Approximation Error.................... 102 4.5.2 Computational Efficiency Expectations................ 103 4.5.3 Identification of Derivative Dependencies............... 105 4.6 Examples.................................... 107 4.6.1 Example 1: Free-Flying Robot Problem............... 109 4.6.2 Example 2: Minimum Time-to-Climb Supersonic Aircraft Problem 111 4.6.3 Example 3: Space Station Attitude Control............. 113 4.7 Discussion................................... 117 4.8 Conclusions................................... 120 5 MESH REFINEMENT METHOD FOR SOLVING BANG-BANG OPTIMAL CONTROL PROBLEMS USING DIRECT COLLOCATION........... 122 5.1 Single-Phase Optimal Control Problem.................... 125 5.2 Legendre-Gauss-Radau Collocation...................... 126 5.3 Control-Linear Hamiltonian.......................... 130 5.4 Bang-Bang Control Mesh Refinement Method................ 132 5.4.1 Method for Identifying Bang-Bang Optimal Control Problems... 132 5.4.2 Estimating Locations of Switches in Control............. 134 5.4.3 Reformulation of Optimal Control Problem Into Multiple Domains. 136 5.4.4 Summary of Bang-Bang Control Mesh Refinement Method..... 138 5.5 Examples.................................... 138 5.5.1 Example 1: Three Compartment Model Problem.......... 140 5.5.2 Example 2: Robot Arm Problem................... 142 6 5.5.3 Example 3: Free-Flying Robot Problem............... 144 5.6 Discussion................................... 147 5.7 Conclusions................................... 148 6 CGPOPS: A C++ SOFTWARE FOR SOLVING MULTIPLE-PHASE OPTIMAL CONTROL PROBLEMS USING ADAPTIVE GAUSSIAN QUADRATURE COLLOCATION AND SPARSE NONLINEAR PROGRAMMING.................................. 149 6.1 General Multiple-Phase Optimal Control Problems............. 151 6.2 Legendre-Gauss-Radau Collocation Method................. 153 6.3 Major Components of CGPOPS ........................ 157 6.3.1 Sparse NLP Arising from Radau Collocation Method........ 157 6.3.1.1 NLP Variables........................ 158 6.3.1.2 NLP Objective and Constraint Functions.......... 159 6.3.2 Sparse Structure of NLP Derivative Functions............ 160 6.3.3 Scaling of Optimal Control Problem for NLP............. 161 6.3.4 Computation Derivatives of NLP Functions.............. 163 6.3.4.1 Central Finite-Difference................... 163 6.3.4.2 Bicomplex-step........................ 164 6.3.4.3 Hyper-Dual.......................... 165 6.3.4.4 Automatic Differentiation.................. 166 6.3.5 Method for Determining the Optimal Control Function Dependencies 166 6.3.6 Adaptive Mesh Refinement....................... 167 6.3.7 Algorithmic Flow of CGPOPS ..................... 168 6.4 Examples.................................... 169 6.4.1 Example 1: Hyper-Sensitive Problem................. 171 6.4.2 Example 2: Reusable Launch Vehicle Entry.............. 175 6.4.3 Example 3: Space Station Attitude Control.............. 179 6.4.4 Example 4: Free-Flying Robot Problem................ 185 6.4.5 Example 5: Multiple-Stage Launch Vehicle Ascent Problem..... 191 6.5 Capabilities of CGPOPS ............................ 195 6.6 Limitations of CGPOPS ............................ 195 6.7 Conclusions................................... 196 7 SUMMARY AND FUTURE WORK........................ 198 REFERENCES....................................... 202 BIOGRAPHICAL SKETCH................................ 212 7 LIST OF TABLES Table page 4-1 Performance results for Example 1 using OC, EC, BC, HD and AD methods for varying number of mesh intervals K using Nk = 5 collocation points in each mesh interval..................................... 111 4-2 Performance results for Example 2 using OC, EC, BC, HD, and AD methods for varying number of mesh intervals K using Nk = 5 in each mesh interval... 114 4-3 Performance results for Example 3 using OC, EC, BC, HD, and AD methods for varying number of mesh intervals K with Nk = 5 collocation points in each mesh interval...................................... 117 5-1 Mesh refinement performance results for Example 1 using hp-BB, hp-I, hp-II, hp-III, and hp-IV mesh refinement methods.................... 142 5-2 Mesh refinement performance results for Example 2 using