Optimal Control with Aerospace Applications Space Technology Library

"Want to know not just what makes rockets go up but how to do it optimally? Optimal control theory has become such an important field in aerospace engineering that no graduate student or practicing engineer can afford to be without a working knowledge of it. This is the first book that begins from scratch to teach the reader the basic principles of the calculus of variations, develop the necessary conditions step-by-step, and introduce the elementary computational techniques of optimal control. This book, with problems and an online solution manual, provides the graduate-level reader with enough introductory knowledge so that he or she can not only read the literature and study the next level textbook but can also apply the theory to find optimal solutions in practice. No more is needed than the usual background of an undergraduate engineering, science, or mathematics program: namely calculus, differential equations, and numerical integration. Although finding optimal solutions for these problems is a complex process involving the calculus of variations, the authors carefully lay out step-by-step the most important theorems and concepts. Numerous examples are worked to demonstrate how to apply the theories to everything from classical problems (e.g., crossing a river in minimum time) to engineering problems (e.g., minimum-fuel launch of a satellite). Throughout the book use is made of the time-optimal launch of a satellite into orbit as an important case study with detailed analysis of two examples: launch from the Moon and launch from Earth. For launching into the field of optimal solutions, look no further!"--Back cover Preface 6 Acknowledgments 8 About the Authors 12 Contents 18 1 Parameter Optimization 22 1.1 Introduction 22 1.2 Parameter Optimization with Constraints 24 1.2.1 Lagrange Multipliers 24 1.2.2 Parameter Optimization: The Hohmann Transfer (1925) 26 1.2.3 Extensions of the Hohmann Transfer (1959) 31 1.2.4 The Bi-parabolic Transfer 33 1.3 Exercises 34 References 38 2 Optimal Control Theory 39 2.1 Optimal Launch of a Satellite 39 2.2 General Form of the Problem 42 2.3 The Problems of Bolza, Lagrange, and Mayer 45 2.3.1 Transformation from Lagrange to Mayer 47 2.3.2 Transformation from Mayer to Lagrange 47 2.4 A Provocative Example Regarding Admissible Functions 48 2.5 Summary 57 2.6 Exercises 57 References 58 3 The Euler-Lagrange Theorem 59 3.1 The Variation 59 3.2 The Euler-Lagrange Equation and the Brachistochrone Problem 61 3.3 The Euler-Lagrange Theorem 65 3.3.1 Proof Outline of the Euler-Lagrange Theorem 66 3.3.2 Summary of the Euler-Lagrange Theorem 72 3.3.3 Alternate Form of the Transversality Condition 72 3.4 Summary 76 3.5 Exercises 77 References 79 4 Application of the Euler-Lagrange Theorem 80 4.1 Introduction 80 4.2 Two-Point Boundary-Value Problem (TPBVP) 80 4.3 Two Approaches to Terminal Constraints 82 4.4 Transversality Condition 84 4.4.1 Case 1 Final Time Specified 84 4.4.2 Case 2 Final State Specified 86 4.4.3 Case 3 Final Endpoint Specified 87 4.5 General Case of Supplying Needed B.C.s 87 4.5.1 Adjoined Method 88 4.5.2 Un-adjoined Method 88 4.6 Examples 89 4.7 A ``Cookbook'' for Optimization Problems 101 4.7.1 Examples of Step 4 103 4.8 Constant Hamiltonian 108 4.9 Summary 109 4.10 Exercises 110 References 113 5 The Weierstrass Condition 114 5.1 Introduction 114 5.2 Statement of the Weierstrass Necessary Condition 114 5.3 Proof Outline of the Weierstrass Necessary Condition 115 5.4 Summary 121 5.5 True or False Quiz for Chaps.1–5 121 References 122 6 The Minimum Principle 123 6.1 Statement of the Minimum Principle 123 6.1.1 Problem Statement 123 6.1.2 Pontryagin's Minimum Principle 124 6.1.3 Examples 125 6.2 Legendre-Clebsch Necessary Condition 130 6.3 Notes on Necessary and Sufficient Conditions 130 6.4 Weak and Strong Extremals 132 6.5 An Example of a Weak but Not Strong Minimum 134 6.6 Second-Order Necessary and Sufficient Conditions 139 6.7 Examples Illustrating the Concept of a Conjugate Point 140 6.8 Summary 145 6.9 Exercises 145 References 146 7 Some Applications 148 7.1 Aircraft Performance Optimization 148 7.2 Maximization of the Range of a Rocket 157 7.2.1 Integration of Equations of Motion When f Is Constant 162 7.2.2 The Optimal Trajectory 163 7.2.3 Maximum Range Equation 164 7.3 Time Optimal Launching of a Satellite 165 7.3.1 Integration of the EOMs 167 7.3.2 TPBVP 173 7.3.3 Flat-Earth Launch Including Atmospheric Drag 174 7.4 Summary 180 7.5 Exercises 180 References 183 8 Weierstrass-Erdmann Corner Conditions 184 8.1 Statement of the Weierstrass-Erdmann Corner Conditions 184 8.2 Proof Outline of Weierstrass-Erdmann Corner Conditions 185 8.3 Summary 190 References 191 9 Bounded Control Problems 192 9.1 Optimal Control Problems with Constraints 192 9.2 Examples of Bounded Control Problems 193 9.3 Singular Arcs 203 9.4 Summary 207 9.5 Exercises 207 References 208 10 General Theory of Optimal Rocket Trajectories 209 10.1 Introduction 209 10.2 Equations of Motion 209 10.3 High and Low-Thrust Engines 210 10.4 Cost Functionals for Rocket Engines 211 10.5 First-Order Necessary Conditions 214 10.5.1 Optimal Constant Specific Impulse Trajectory 214 10.5.2 Optimal Impulsive Trajectory 218 10.5.3 Optimal Variable Specific Impulse Trajectory 220 10.6 Optimal Trajectories in a Uniform Field 222 10.7 Summary 224 10.8 Exercises 225 10.9 True or False Quiz for Chaps.6–10 226 References 228 Appendices 229 A Time-Optimal Lunar Ascent 230 A.1 MATLAB's Two-Point Boundary-Value Solver 230 A.2 Solution Method 231 A.3 MATLAB Code 232 B Time-Optimal Launch of a Titan II 237 B.1 Scaling the TPBVP 237 B.2 Solution Method 241 B.3 Results 241 B.4 MATLAB Code 243 C Optimal Low-Thrust LEO to GEO Circular Orbit Transfer 249 C.1 Optimization Problem 249 C.2 Scaling the Equations of Motion 250 C.3 Applying the Euler-Lagrange Theorem 252 C.4 Boundary Conditions and the TPBVP 253 C.5 Results 255 Discussion of Results 256 C.6 MATLAB Code 257 Executable File 257 Reference 261 D Curious Quotations 262 Bibliography (Books) 264 Bibliography (Aerospace Applications Papers and Reports) 269 Index 282 Front Matter....Pages i-xx Parameter Optimization....Pages 1-17 Optimal Control Theory....Pages 19-38 The Euler-Lagrange Theorem....Pages 39-59 Application of the Euler-Lagrange Theorem....Pages 61-94 The Weierstrass Condition....Pages 95-103 The Minimum Principle....Pages 105-129 Some Applications....Pages 131-166 Weierstrass-Erdmann Corner Conditions....Pages 167-174 Bounded Control Problems....Pages 175-191 General Theory of Optimal Rocket Trajectories....Pages 193-212 Back Matter....Pages 213-273