Opis
Introduction to the Mathematical Theory of Control Processes
Zwiń wyszukiwarkę
(z VAT)
Cena netto: PLN
Forma zakupu
Do koszyka
Introduction to the Mathematical Theory of Control Processes
Linear Equations and Quadratic Criteria
Autorzy: Richard Bellman Wydawnictwo: Elsevier Science Data wydania: 2016 Język publikacji: Angielski Liczba stron: 264 Formaty publikacji: EAN: 9781483222868 ISBN: 9781483222868 Kategoria: Mathematics Indeks wydawcy: 9781483222868 Nota bibliograficzna: -
Spis treści
- Front Cover 2
- Linear Equations and Quadratic Criteria 7
- Copyright Page 8
- INTRODUCTION 9
- Table of Contents 13
-
Chapter 1. WHAT IS CONTROL THEORY?
21
- 1.1. Introduction 21
- 1.2. Systems 22
- 1.3. Schematics 22
- 1.5. The Behavior of Systems 24
- 1.4. Mathematical Systems 24
- 1.6. Improvement of the Behavior of Systems 25
- 1.7. More Detailed Breakdown 26
- 1.8. Uncertainty 27
- 1.9. Conclusion 27
- Front Cover 2
- Linear Equations and Quadratic Criteria 7
- Copyright Page 8
- INTRODUCTION 9
- Table of Contents 13
-
Chapter 1. WHAT IS CONTROL THEORY?
21
- 1.1. Introduction 21
- 1.2. Systems 22
- 1.3. Schematics 22
- 1.5. The Behavior of Systems 24
- 1.4. Mathematical Systems 24
- 1.6. Improvement of the Behavior of Systems 25
- 1.7. More Detailed Breakdown 26
- 1.8. Uncertainty 27
- 1.9. Conclusion 27
- BIBLIOGRAPHY AND COMMENTS 28
-
Chapter 2. SECOND-ORDER UNEAR DIFFERENTIAL AND DIFFERENCE EQUATIONS
30
- 2.2. Second-Order Linear Differential Equations with Constant Coefficients 30
- 2.1. Introduction 30
- 2.3. The Inhomogeneous Equation 32
- 2.4. Two-Point Boundary Conditions 33
- 2.5. First-Order Linear Differential Equations with Variable Coefficients 34
- 2.6. The Riccati Equation 35
- 2.7. Linear Equations with Variable Coefficients 37
- 2.8. The Inhomogeneous Equation 38
- 2.9. Green's Function 40
- 2.10. Linear Systems 42
- 2.11. Difference Equations 43
- Miscellaneous Exercises 45
- BIBLIOGRAPHY AND COMMENTS 47
-
Chapter 3. STABILITY AND CONTROL
48
- 3.1 Introduction 48
- 3.2. Stability 50
- 3.3. Numerical Solution and Stability 53
- 3.4. Perturbation Procedures 54
- 3.5. A Fundamental Stability Theorem 56
- 3.7. Stability by Control 57
- 3.6. Stability by Design 57
- 3.8. Proportional Control 59
- 3.9. Discussion 60
- 3.10. Analytic Formulation 61
- 3.11. One-Dimensional Systems 62
- Miscellaneous Exercises 64
- BIBLIOGRAPHY AND COMMENTS 66
-
Chapter 4. CONTINUOUS VARIATIONAL PROCESSES; CAICULUS OF VARIATIONS
69
- 4.1. Introduction 69
- 4.2. Does a Minimum Exist? 70
- 4.3. The Euler Equation 72
- 4.4. A Fallacious Argument 74
- 4.5. Haar's Device 74
- 4.6. Solution of the Euler Equation 75
- 4.7. Minimizing Property of the Solution 76
- 4.9. Asymptotic Control 77
- 4.8. Alternative Approach 77
- 4.10. Infinite Control Process 79
- 4.11. The Minimum Value of J(u) 80
- 4.12. Two-Point Constraints 82
- 4.13. Terminal Control 83
- 4.14. The Courant Parameter 85
- 4.15. Successive Approximations 85
- 4.16. min ∫t0 [u2 + g(t)u2] dt 87
- 4.17. Discussion 89
- 4.18. The Simplicity of Control Processes 89
- 4.19. Discussion 91
- 4.20. The Minimum Value of j(u) 92
- 4.21. A Smoothing Process 92
- 4.22. Variation-Diminishing Property of Green's Function 94
- 4.23. Constraints 96
- 4.25. Monotonicity in λ 97
- 4.24. Minimizing Property 97
- 4.26. Proof of Monotonicity 98
- 4.27. Discussion 99
- 4.28. More General Quadratic Variational Problems 99
- 4.29. Variational Procedure 100
- 4.30. Proof of Minimum Property 101
- 4.31. Existence and Uniqueness 102
- 4.32. The Adjoint Operator 102
- 4.33. Sturm-Liouville Theory 105
- 4.34. Minimization by Means of Inequalities 107
- 4.35. Multiple Constraints 108
- 4.36. Unknown External Forces 110
- Miscellaneous Exercises 111
- BIBLIOGRAPHY AND COMMENTS 119
-
Chapter 5. DYNAMIC PROGRAMMING
121
- 5.2. Control as a Multistage Decision Process 121
- 5.1. Introduction 121
- 5.3. Preliminary Concepts 123
- 5.4. Formalism 124
- 5.5. Principle of Optimality 127
- 5.6. Discussion 128
- 5.7. Simplification 129
- 5.8. Validation 129
- 5.10. Limiting Behavior as T→∞ 130
- 5.9. Infinite Process 130
- 5.11. Two-Point Boundary Problems 132
- 5.12. Time-Dependent Control Process 133
- 5.13. Global Constraints 134
- 5.14. Discrete Control Processes 136
- 5.15. Preliminaries 137
- 5.16. Recurrence Relation 137
- 5.17. Explicit Recurrence Relations 138
- 5.18. Behavior of rN 139
- 5.20. Equivalent Linear Relations 140
- 5.19. Approach to Steady-State Behavior 140
- 5.21. Local Constraints 141
- 5.22. Continuous as Limit of Discrete 143
- 5.23. Bang-Bang Control 144
- 5.24. Control in the Presence of Unknown Influences 145
- 5.25. Comparison between Calculus of Variations and Dynamic Programming 146
- Miscellaneous Exercises 147
- BIBLIOGRAPHY AND COMMENTS 155
-
Chapter 6. REVIEW OF MATRIX THEORY AND LINEAR DIFFERENTIAL EQUATIONS
158
- 6.1. Introduction 158
- 6.2. Vector-Matrix Notation 159
- 6.3. Inverse Matrix 160
- 6.4. The Product of Two Matrices 161
- 6.5. Inner Product and Norms 163
- 6.6. Orthogonal Matrices 165
- 6.7. Canonical Representation 166
- 6.8. Det A 167
- 6.9. Functions of a Symmetric Matrix 167
- 6.10. Positive Definite Matrices 169
- 6.11. Representation of A 170
- 6.12. Differentiation and Integration of Vectors and Matrices 171
- 6.14. Existence and Uniqueness Proof 172
- 6.13. The Matrix Exponential 172
- 6.15. Euler Technique and Asymptotic Behavior 176
- 6.16. x" - A(t)x = 0 177
- 6.17. x' = Ax + By, y' = Cx + Dy 178
- 6.19. dX/dt = AX + XB, X(0) = C 179
- 6.18. Matrix Riccati Equation 179
- Miscellaneous Exercises 180
- BIBLIOGRAPHY AND COMMENTS 184
-
Chapter 7. MULTIDIMENSIONAL CONTROL PROCESSES VIA THE CALCULUS OF VARIATIONS
185
- 7.1. Introduction 185
- 7.2. The Euler Equation 186
- 7.3. The Case of Constant A 187
- 7.5. The Minimum Value 189
- 7.4. Nonsingularity of cosh ST 189
- 7.6. Asymptotic Behavior 190
- 7.8. The Nonsingularity of X2'(T) 191
- 7.7. Variable A(t) 191
- 7.9. The Minimum Value 192
- 7.10. Computational Aspects 193
- 7.11. min ∫T0 [(x, x) + (y, y)] dt, x' = Bx + y, x(0) = c 194
- Miscellaneous Exercises 195
-
Chapter 8. MULTIDIMENSIONAL CONTROL PROCESSES VIA DYNAMIC PROGRAMMING
198
- 8.1. Introduction 198
- 8.2. ∫T0 [(x', x') + (x, Ax)] dt 199
- 8.3. The Associated Riccati Equation 200
- 8.4. Asymptotic Behavior 201
- 8.5. Rigorous Aspects 202
- 8.6. Time-Dependent Case 202
- 8.7. Computational Aspects 203
- 8.8. Successive Approximations 204
- 8.9. Approximation in Policy Space 205
- 8.10. Monotone Convergence 206
- 8.11. Partitioning 207
- 8.12. Power Series Expansions 208
- 8.13. Extrapolation 209
- 8.14. Minimization via Inequalities 211
- 8.15. Discrete Control Processes 213
- 8.16. Ill-Conditioned Linear Systems 215
- 8.17. Lagrange Multipliers 218
- 8.18. Reduction of Dimensionality 219
- 8.19. Successive Approximations 222
- 8.20. Distributed Parameters 224
- 8.21. Slightly Intertwined Systems 225
- Miscellaneous Exercises 228
- BIBLIOGRAPHY AND COMMENTS 234
-
Chapter 9. FUNCTIONAL ANALYSIS
237
- 9.1. Motivation 237
- 9.2. The Hilbert Space L2(0, T) 238
- 9.3. Inner Products 239
- 9.5. Vector Hilbert Space 240
- 9.4. Linear Operators 240
- 9.6. Quadratic Functionals 241
- 9.7. Existence and Uniqueness of a Minimizing Function 242
- 9.8. The Equation for the Minimizing Function 244
- 9.9. Application to Differential Equations 246
- 9.10. Numerical Aspects 247
- 9.11. A Simple Algebraic Example 248
- 9.12. The Equation x + λBx = c 249
- 9.13. The Integral Equation ƒ(t) + λ∫T0 K(t, t1)ƒ(t1) dt1 = g(t) 251
- 9.14. Lagrange Multipliers 251
- 9.15. The Operator Ra 252
- 9.16. Control Subject to Constraints 254
- 9.17. Properties of φ(c) and Ψ(c) 256
- 9.18. Statement of Result 256
- Miscellaneous Exercises 257
- BIBLIOGRAPHY AND COMMENTS 260
- AUTHOR INDEX 261
- SUBJECT INDEX 263