Computing Methods in Optimization Problems (eBook)

Front Cover 1
Computing Methods in Optimization Problems 4
Copyright Page 5
Table of Contents 10
Preface 6
LIST OF CONTRIBUTORS 8
CHAPTER 1. VARIATIONAL THEORY AND OPTIMAL CONTROL THEORY 12
1. Introduction 12
2. First-order necessary conditions for a local minimum 16
3. Further remarks relative to constraints 18
4. A basic lemma 19
5. Control problems without state variables 21
6. Control problems linear in the state variables 24
7. Control problem nonlinear in the state variables 27
8. Second-order necessary conditions 30
REFERENCES 32
CHAPTER 2. ON THE COMPUTATION OF THE OPTIMAL TEMPERATURE PROFILE IN A TUBULAR REACTION VESSEL 34
1. THE PROBLEM AND ITS MATHEMATICAL FORMULATION 36
2. BEST ISOTHERMAL YIELD 38
3. FIRST METHOD OF SOLUTION: DISCRETE APPROXIMATION 40
4. SECOND METHOD OF SOLUTION: PARAMETRIC EXPANSION 41
5. THIRD METHOD OF SOLUTION: PONTRYAGIN'S MAXIMUM PRINCIPLE 47
6. FOURTH METHOD OF SOLUTION: GRADIENT METHOD IN FUNCTION SPACE 51
7. FIFTH METHOD OF SOLUTION: DYNAMIC PROGRAMMING 60
8. CONCLUSIONS 66
APPENDIX I: THE TRAPEZOIDAL RULE FOR INTEGRATION 68
APPENDIX 2: DESCRIPTION AND COMPARISON OF TECHNIQUES 69
REFERENCES 74
CHAPTER 3. SEVERAL TRAJECTORY OPTIMIZATION TECHNIQUES 76
Introduction 76
Problem Formulation 77
Gradient Techniques 82
Second Variation Method 88
Generalized Newton-Raphson Method 93
Conclusions 96
Acknowledgments 98
References 99
CHAPTER 4. SEVERAL TRAJECTORY OPTIMIZATION TECHNIQUES 102
Introduction and Statement of Problem 102
The Three Methods as Applied to the Sample Problem 103
Acknowledgments 114
References 114
CHAPTER 5. A STEEPEST-ASCENT TRAJECTORY OPTIMIZATION METHOD WHICH REDUCES MEMORY REQUIREMENTS 118
1. Adjoint Equations 119
2. Necessary Conditions and the Hamiltonian 122
3. Steepest-Ascent Method 124
4. Discussion 133
5. Computation Review and Glossary 134
6. Initiation Procedure 138
ACKNOWLEDGMENT 144
CHAPTER 6. DYNAMIC PROGRAMMING, INVARIANT IMBEDDING AND QUASILINEARIZATION: COMPARISONS AND INTERCONNECTIONS 146
1. Introduction 146
2. Quasilinearization 146
3. Dynamic Programming 148
4. Invariant Imbedding 149
5. Combined Calculations 152
References 154
CHAPTER 7. A COMPARISON BETWEEN SOME METHODS FOR COMPUTING OPTIMUM PATHS IN THE PROBLEM OF BOLZA 158
1. Basic Equations 160
2. Method of the Fundamental Lemma 162
3. Differential Method 163
4. Determining a 'Best-fitting' Extremal 164
5. Comments 165
References 167
CHAPTER 8. MINIMIZING FUNCTIONALS ON HILBERT SPACE 170
Acknowledgments 175
References 175
CHAPTER 9. COMPUTATIONAL ASPECTS OF THE TIME-OPTIMAL CONTROL PROBLEM 178
1. Introduction 178
2. The Time Optimal-Control Problem 178
3. The Computation of Optimal Controls 181
4. The Linearized Difference Equation 183
5. Example Problem 187
6. Methods for Improving Convergence 191
7. The Derivation of H(n) 195
8. Extensions 198
Bibliography 199
Notes for Tables 201
CHAPTER 10. AN ON-LINE IDENTIFICATION SCHEME for MULTIVARIABLE NONLINEAR SYSTEMS 204
I. INTRODUCTION 204
II. DESCRIPTION OF THE MULTIVARIATES LE NONLINEAR SYSTEMS 206
III. FORMULATION OF THE IDENTIFICATION PROBLEM FOR NONLINEAR SYSTEMS 209
IV. SOLUTION OF THE IDENTIFICATION PROBLEM BY THE STEEPEST DESCENT METHOD IN THE ADJOINT SPACE 210
V. RECURSIVE ESTIMATION OF SYSTEM WEIGHTING FUNCTION MATRICES FOR GROWING DATA 213
VI. RECURSIVE ESTIMATION OF SYSTEM WEIGHTING FUNCTION MATRIX FOR FIXED DATA LENGHT 217
REFERENCES 218
CHAPTER 11. METHOD OF CONVEX ASCENT 222
Introduction 222
Section 1. Statement of the Problem 229
Section 2. Comoving Coordinate Space Along a Given Trajectory 231
Some algebraic manipulations 233
Section 3. Reachable Sets 236
Section 4. Necessary Condition for the Optimal Control of Nonlinear Systems 237
Comments on the logical structure of the previous theorems 237
Section 5. Optimal Solution for the V-Approximate Differential Equation 238
Important Remark 240
Section 6. Optimal Gain of the Iterative Computational Procedure 242
Section 7. Application of the Method of Convex Ascent to the Goddard Problem 245
Acknowledgments 247
REFERENCES 248
CHAPTER 12. STUDY OF AN ALGORITHM FOR DYNAMIC OPTIMIZATION 252
I. INTRODUCTION 252
II. AUTOSTABILITY 253
III. DYNAMIC TRAJECTORY 254
IV. NATURAL SWITCHING 256
V. TIME-LAG COMPENSATION 257
VI. FREE OSCILLATIONS 257
VII. NF CYCLES 257
VIII. NN CYCLES 263
IX. FF CYCLES 268
X. FORCED OSCILLATIONS 268
XI. CONCLUSIONS 268
ACKNOWLEDGMENTS 270
CHAPTER 13. THE APPLICATION OF HYBRID COMPUTERS TO THE ITERATIVE SOLUTION OF OPTIMAL CONTROL PROBLEMS 272
1. Introduction 272
2. Iterative Solution of the Time-Optimal Control Problem 272
3. Modified Formulation of the Iterative Procedure 275
4. The Hybrid Computer 277
5. The Programming of the Iterative Procedure 280
6. Example Computations 289
7. Conclusions and Acknowledgements 294
References 294
CHAPTER 14. SYNTHESIS OF OPTIMAL CONTROLLERS USING HYBRID ANALOG-DIGITAL COMPUTERS 296
I. Introduction 296
II. Computer Studies 297
III. HYDAC Simulation 304
Appendix A 307
Appendix B 308
CHAPTER 15. GRADIENT METHODS FOR THE OPTIMIZATION OF DYNAMIC SYSTEM PARAMETERS BY HYBRID COMPUTATION 316
I. INTRODUCTION 316
2. CONTINUOUS PARAMETER OPTIMIZATION 317
3. DISCRETE PARAMETER OPTIMIZATION 325
4. HYBRID COMPUTER IMPLEMENTATION 331
5. CONCLUSIONS 336
APPENDIX 336
REFERENCES 337