Balakrishnan / Neustadt Computing Methods in Optimization Problems

Proceedings of a Conference Held at University of California, Los Angeles January 30-31, 1964
1. Auflage 2014
ISBN: 978-1-4832-2315-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 338 Seiten, Web PDF

Computing Methods in Optimization Problems deals with hybrid computing methods and optimization techniques using computers. One paper discusses different numerical approaches to optimizing trajectories, including the gradient method, the second variation method, and a generalized Newton-Raphson method. The paper cites the advantages and disadvantages of each method, and compares the second variation method (a direct method) with the generalized Newton-Raphson method (an indirect method). An example problem illustrates the application of the three methods in minimizing the transfer time of a low-thrust ion rocket between the orbits of Earth and Mars. Another paper discusses an iterative process for steepest-ascent optimization of orbit transfer trajectories to minimize storage requirements such as in reduced memory space utilized in guidance computers. By eliminating state variable storage and control schedule storage, the investigator can achieve reduced memory requirements. Other papers discuss dynamic programming, invariant imbedding, quasilinearization, Hilbert space, and the computational aspects of a time-optimal control problem. The collection is suitable for computer programmers, engineers, designers of industrial processes, and researchers involved in aviation or control systems technology.

Autoren/Hrsg.

Balakrishnan, A. V.

Neustadt, Lucien W.

Inhaltsverzeichnis

1;Front Cover;1
2;Computing Methods in Optimization Problems;4
3;Copyright Page;5
4;Table of Contents;10
5;Preface;6
6;LIST OF CONTRIBUTORS;8
7;CHAPTER 1. VARIATIONAL THEORY AND OPTIMAL CONTROL THEORY;12
7.1;1. Introduction;12
7.2;2. First-order necessary conditions for a local minimum;16
7.3;3. Further remarks relative to constraints;18
7.4;4. A basic lemma;19
7.5;5. Control problems without state variables;21
7.6;6. Control problems linear in the state variables;24
7.7;7. Control problem nonlinear in the state variables;27
7.8;8. Second-order necessary conditions;30
7.9;REFERENCES;32
8;CHAPTER 2. ON THE COMPUTATION OF THE OPTIMAL TEMPERATURE PROFILE IN A TUBULAR REACTION VESSEL;34
8.1;1. THE PROBLEM AND ITS MATHEMATICAL FORMULATION;36
8.2;2. BEST ISOTHERMAL YIELD;38
8.3;3. FIRST METHOD OF SOLUTION: DISCRETE APPROXIMATION;40
8.4;4. SECOND METHOD OF SOLUTION: PARAMETRIC EXPANSION;41
8.5;5. THIRD METHOD OF SOLUTION: PONTRYAGIN'S MAXIMUM PRINCIPLE;47
8.6;6. FOURTH METHOD OF SOLUTION: GRADIENT METHOD IN FUNCTION SPACE;51
8.7;7. FIFTH METHOD OF SOLUTION: DYNAMIC PROGRAMMING;60
8.8;8. CONCLUSIONS;66
8.9;APPENDIX I: THE TRAPEZOIDAL RULE FOR INTEGRATION;68
8.10;APPENDIX 2: DESCRIPTION AND COMPARISON OF TECHNIQUES;69
8.11;REFERENCES;74
9;CHAPTER 3. SEVERAL TRAJECTORY OPTIMIZATION TECHNIQUES;76
9.1;Introduction;76
9.2;Problem Formulation;77
9.3;Gradient Techniques;82
9.4;Second Variation Method;88
9.5;Generalized Newton-Raphson Method;93
9.6;Conclusions;96
9.7;Acknowledgments;98
9.8;References;99
10;CHAPTER 4. SEVERAL TRAJECTORY OPTIMIZATION TECHNIQUES;102
10.1;Introduction and Statement of Problem;102
10.2;The Three Methods as Applied to the Sample Problem;103
10.3;Acknowledgments;114
10.4;References;114
11;CHAPTER 5. A STEEPEST-ASCENT TRAJECTORY OPTIMIZATION METHOD WHICH REDUCES MEMORY REQUIREMENTS;118
11.1;1. Adjoint Equations;119
11.2;2. Necessary Conditions and the Hamiltonian;122
11.3;3. Steepest-Ascent Method;124
11.4;4. Discussion;133
11.5;5. Computation Review and Glossary;134
11.6;6. Initiation Procedure;138
11.7;ACKNOWLEDGMENT;144
12;CHAPTER 6. DYNAMIC PROGRAMMING, INVARIANT IMBEDDING AND QUASILINEARIZATION: COMPARISONS AND INTERCONNECTIONS;146
12.1;1. Introduction;146
12.2;2. Quasilinearization;146
12.3;3. Dynamic Programming;148
12.4;4. Invariant Imbedding;149
12.5;5. Combined Calculations;152
12.6;References;154
13;CHAPTER 7. A COMPARISON BETWEEN SOME METHODS FOR COMPUTING OPTIMUM PATHS IN THE PROBLEM OF BOLZA;158
13.1;1. Basic Equations;160
13.2;2. Method of the Fundamental Lemma;162
13.3;3. Differential Method;163
13.4;4. Determining a 'Best-fitting' Extremal;164
13.5;5. Comments;165
13.6;References;167
14;CHAPTER 8. MINIMIZING FUNCTIONALS ON HILBERT SPACE;170
14.1;Acknowledgments;175
14.2;References;175
15;CHAPTER 9. COMPUTATIONAL ASPECTS OF THE TIME-OPTIMAL CONTROL PROBLEM;178
15.1;1. Introduction;178
15.2;2. The Time Optimal-Control Problem;178
15.3;3. The Computation of Optimal Controls;181
15.4;4. The Linearized Difference Equation;183
15.5;5. Example Problem;187
15.6;6. Methods for Improving Convergence;191
15.7;7. The Derivation of H(n);195
15.8;8. Extensions;198
15.9;Bibliography;199
15.10;Notes for Tables;201
16;CHAPTER 10. AN ON-LINE IDENTIFICATION SCHEME for MULTIVARIABLE NONLINEAR SYSTEMS;204
16.1;I. INTRODUCTION;204
16.2;II. DESCRIPTION OF THE MULTIVARIATES LE NONLINEAR SYSTEMS;206
16.3;III. FORMULATION OF THE IDENTIFICATION PROBLEM FOR NONLINEAR SYSTEMS;209
16.4;IV. SOLUTION OF THE IDENTIFICATION PROBLEM BY THE STEEPEST DESCENT METHOD IN THE ADJOINT SPACE;210
16.5;V. RECURSIVE ESTIMATION OF SYSTEM WEIGHTING FUNCTION MATRICES FOR GROWING DATA;213
16.6;VI. RECURSIVE ESTIMATION OF SYSTEM WEIGHTING FUNCTION MATRIX FOR FIXED DATA LENGHT;217
16.7;REFERENCES;218
17;CHAPTER 11. METHOD OF CONVEX ASCENT;222
17.1;Introduction;222
17.2;Section 1. Statement of the Problem;229
17.3;Section 2. Comoving Coordinate Space Along a Given Trajectory;231
17.4;Some algebraic manipulations;233
17.5;Section 3. Reachable Sets;236
17.6;Section 4. Necessary Condition for the Optimal Control of Nonlinear Systems;237
17.7;Comments on the logical structure of the previous theorems;237
17.8;Section 5. Optimal Solution for the V-Approximate Differential Equation;238
17.9;Important Remark;240
17.10;Section 6. Optimal Gain of the Iterative Computational Procedure;242
17.11;Section 7. Application of the Method of Convex Ascent to the Goddard Problem;245
17.12;Acknowledgments;247
17.13;REFERENCES;248
18;CHAPTER 12. STUDY OF AN ALGORITHM FOR DYNAMIC OPTIMIZATION;252
18.1;I. INTRODUCTION;252
18.2;II. AUTOSTABILITY;253
18.3;III. DYNAMIC TRAJECTORY;254
18.4;IV. NATURAL SWITCHING;256
18.5;V. TIME-LAG COMPENSATION;257
18.6;VI. FREE OSCILLATIONS;257
18.7;VII. NF CYCLES;257
18.8;VIII. NN CYCLES;263
18.9;IX. FF CYCLES;268
18.10;X. FORCED OSCILLATIONS;268
18.11;XI. CONCLUSIONS;268
18.12;ACKNOWLEDGMENTS;270
19;CHAPTER 13. THE APPLICATION OF HYBRID COMPUTERS TO THE ITERATIVE SOLUTION OF OPTIMAL CONTROL PROBLEMS;272
19.1;1. Introduction;272
19.2;2. Iterative Solution of the Time-Optimal Control Problem;272
19.3;3. Modified Formulation of the Iterative Procedure;275
19.4;4. The Hybrid Computer;277
19.5;5. The Programming of the Iterative Procedure;280
19.6;6. Example Computations;289
19.7;7. Conclusions and Acknowledgements;294
19.8;References;294
20;CHAPTER 14. SYNTHESIS OF OPTIMAL CONTROLLERS USING HYBRID ANALOG-DIGITAL COMPUTERS;296
20.1;I. Introduction;296
20.2;II. Computer Studies;297
20.3;III. HYDAC Simulation;304
20.4;Appendix A;307
20.5;Appendix B;308
21;CHAPTER 15. GRADIENT METHODS FOR THE OPTIMIZATION OF DYNAMIC SYSTEM PARAMETERS BY HYBRID COMPUTATION;316
21.1;I. INTRODUCTION;316
21.2;2. CONTINUOUS PARAMETER OPTIMIZATION;317
21.3;3. DISCRETE PARAMETER OPTIMIZATION;325
21.4;4. HYBRID COMPUTER IMPLEMENTATION;331
21.5;5. CONCLUSIONS;336
21.6;APPENDIX;336
21.7;REFERENCES;337