The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ordinary differential equations and differential-algebraic equations. An emphasis is placed on the interplay between the continuous optimal control problem, which typically is defined and analyzed in a Banach space setting, and discrete optimal control problems, which are obtained by discretization and lead to finite dimensional optimization problems.
Author(s): Matthias Gerdts
Series: de Gruyter Textbook
Publisher: de Gruyter
Year: 2011
Language: English
Pages: 458
Tags: Автоматизация;Теория автоматического управления (ТАУ);Книги на иностранных языках;
Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
1 Introduction......Page 12
1.1 DAE Optimal Control Problems......Page 19
1.1.1 Perturbation Index......Page 35
1.1.2 Consistent Initial Values......Page 41
1.1.3 Index Reduction and Stabilization......Page 43
1.2.1 Transformation to Fixed Time Interval......Page 47
1.2.2 Transformation to Autonomous Problem......Page 48
1.2.4 Transformation of L1-Minimization Problems......Page 49
1.2.5 Transformation of Interior-Point Constraints......Page 50
1.3 Overview......Page 53
1.4 Exercises......Page 55
2.1 Function Spaces......Page 61
2.1.1 Topological Spaces, Banach Spaces, and Hilbert Spaces......Page 62
2.1.2 Mappings and Dual Spaces......Page 65
2.1.3 Derivatives, Mean-Value Theorem, and Implicit Function Theorem......Page 67
2.1.4 Lp-Spaces, Wq,p-Spaces, Absolutely Continuous Functions, Functions of Bounded Variation......Page 71
2.2 The DAE Optimal Control Problem as an Infinite Optimization Problem......Page 79
2.3 Necessary Conditions for Infinite Optimization Problems......Page 86
2.3.1 Existence of a Solution......Page 88
2.3.2 Conic Approximation of Sets......Page 90
2.3.3 Separation Theorems......Page 95
2.3.4 First Order Necessary Optimality Conditions of Fritz John Type......Page 98
2.3.5 Constraint Qualifications and Karush-Kuhn-Tucker Conditions......Page 106
2.4 Exercises......Page 111
3 Local Minimum Principles......Page 115
3.1 Problems without Pure State and Mixed Control-State Constraints......Page 117
3.1.1 Representation of Multipliers......Page 122
3.1.2 Local Minimum Principle......Page 125
3.1.3 Constraint Qualifications and Regularity......Page 129
3.2 Problems with Pure State Constraints......Page 136
3.2.1 Representation of Multipliers......Page 138
3.2.2 Local Minimum Principle......Page 141
3.2.3 Finding Controls on Active State Constraint Arcs......Page 146
3.2.4 Jump Conditions for the Adjoint......Page 148
3.3 Problems with Mixed Control-State Constraints......Page 152
3.3.1 Representation of Multipliers......Page 154
3.3.2 Local Minimum Principle......Page 156
3.4 Summary of Local Minimum Principles for Index-One Problems......Page 159
3.5 Exercises......Page 163
4 Discretization Methods for ODEs and DAEs......Page 168
4.1.1 The Euler Method......Page 170
4.1.2 Runge-Kutta Methods......Page 173
4.1.4 Consistency, Stability, and Convergence of One-Step Methods......Page 179
4.2 Backward Differentiation Formulas (BDF)......Page 187
4.3 Linearized Implicit Runge-Kutta Methods......Page 189
4.4 Automatic Step-size Selection......Page 196
4.5 Computation of Consistent Initial Values......Page 202
4.5.1 Projection Method for Consistent Initial Values......Page 203
4.5.2 Consistent Initial Values via Relaxation......Page 204
4.6 Shooting Techniques for Boundary Value Problems......Page 206
4.6.1 Single Shooting Method using Projections......Page 207
4.6.2 Single Shooting Method using Relaxations......Page 214
4.6.3 Multiple Shooting Method......Page 215
4.7 Exercises......Page 219
5 Discretization of Optimal Control Problems......Page 226
5.1 Direct Discretization Methods......Page 227
5.1.1 Full Discretization Approach......Page 229
5.1.2 Reduced Discretization Approach......Page 231
5.1.3 Control Discretization......Page 233
5.2.1 Lagrange-Newton Method......Page 238
5.2.2 Sequential Quadratic Programming (SQP)......Page 240
5.3 Calculation of Derivatives for Reduced Discretization......Page 249
5.3.1 Sensitivity Equation Approach......Page 250
5.3.2 Adjoint Equation Approach: The Discrete Case......Page 252
5.3.3 Adjoint Equation Approach : The Continuous Case......Page 259
5.4 Discrete Minimum Principle and Approximation of Adjoints......Page 265
5.4.1 Example......Page 273
5.5.1 Convergence of the Euler Discretization......Page 284
5.5.2 Higher Order of Convergence for Runge-Kutta Discretizations......Page 287
5.6 Numerical Examples......Page 289
5.7 Exercises......Page 299
6 Real-Time Control......Page 303
6.1.1 Parametric Sensitivity Analysis of Nonlinear Optimization Problems......Page 304
6.1.2 Open-Loop Real-Time Control via Sensitivity Analysis......Page 314
6.2 Feedback Controller Design by Optimal Control Techniques......Page 326
6.3 Model Predictive Control......Page 336
6.4 Exercises......Page 343
7 Mixed-Integer Optimal Control......Page 349
7.1 Global Minimum Principle......Page 350
7.1.1 Singular Controls......Page 362
7.2 Variable Time Transformation Method......Page 370
7.3 Switching Costs, Dynamic Programming, Bellman’s Optimality Principle......Page 384
7.3.1 Dynamic Optimization Model with Switching Costs......Page 385
7.3.2 A Dynamic Programming Approach......Page 387
7.3.3 Examples......Page 394
7.4 Exercises......Page 402
8 Function Space Methods......Page 405
8.1 Gradient Method......Page 406
8.2 Lagrange-Newton Method......Page 422
8.2.1 Computation of the Search Direction......Page 428
8.3 Exercises......Page 440
Bibliography......Page 444
Index......Page 466
