This textbook is the outgrowth of teaching analytical optimization to aerospace engineering graduate students. To make the material available to the widest audience, the prerequisites are limited to calculus and differential equations. It is also a book about the mathematical aspects of optimal control theory. It was developed in an engineering environment from material learned by the author while applying it to the solution of engineering problems. One goal of the book is to help engineering graduate students learn the fundamentals which are needed to apply the methods to engineering problems. The examples are from geometry and elementary dynamical systems so that they can be understood by all engineering students. Another goal of this text is to unify optimization by using the differential of calculus to create the Taylor series expansions needed to derive the optimality conditions of optimal control theory.
Author(s): David G. Hull
Series: Mechanical Engineering Series
Publisher: Springer
Year: 2003
Language: English
Pages: 402
Tags: Автоматизация;Теория автоматического управления (ТАУ);Книги на иностранных языках;
Front cover......Page 1
Title page......Page 3
Copyright page......Page 4
Dedication......Page 5
Series preface......Page 6
Preface......Page 7
Contents......Page 11
1.2 Classification of Systems......Page 21
1.3.1 Distance Problem......Page 22
1.3.2 General Parameter Optimization Problem......Page 23
1.4 Optimal Control Theory......Page 24
1.4.1 Distance Problem......Page 25
1.4.2 Acceleration Problem......Page 26
1.4.3 Navigation Problem......Page 28
1.4.4 General Optimal Control Problem......Page 31
1.4.5 Conversion of an Optimal Control Problem into a Parameter Optimization Problem......Page 33
1.6 Text Organization......Page 34
Part I. Parameter Optimization......Page 36
2.2 Taylor Series and Differentials......Page 38
2.3 Function of One Variable......Page 42
2.3.2 Second Differential Conditions......Page 46
2.3.4 Summary......Page 47
2.4 Distance Problem......Page 49
2.5.1 First Differential Conditions......Page 50
2.5.2 Second Differential Conditions......Page 51
2.5.3 Summary......Page 54
2.6 Examples......Page 55
2.7 Historical Example......Page 56
2.8 Function of $n$ Independent Variables......Page 58
3.1 Introduction......Page 62
3.2 Function of Two Constrained Variables......Page 63
3.2.1 Direct Approach......Page 64
3.2.2 Lagrange Multiplier Approach......Page 65
3.2.3 Remarks......Page 67
3.3 Distance Problem......Page 70
3.4 Function of $n$ Constrained Variables......Page 72
3.5 Example......Page 73
4.1 Introduction......Page 79
4.2 Boundary Minimal Points......Page 80
4.3 Introduction to Slack Variables......Page 81
4.4 Function of Two Variables......Page 83
4.5 Example......Page 86
4.6 Eliminating Bounded Variables......Page 87
4.7 Linear Programming Examples......Page 88
4.8 General Problem......Page 91
5.2 Matrix Algebra......Page 96
5.2.2 Multiplication......Page 97
5.3.1 Differential......Page 98
5.3.2 Integration......Page 101
5.5 Function of $n$ Constrained Variables......Page 102
Part II. Optimal Control Theory......Page 105
6.1 Introduction......Page 108
6.2 Standard Optimal Control Problem......Page 109
6.3 Differential of the State Equation......Page 111
6.4 Relationship Between $\delta$ and $d$......Page 115
6.5 Differential of the Final Condition......Page 117
6.6 Differential of the Integral......Page 118
6.7 Summary of Differential Properties......Page 120
7.2 Fixed Final Time......Page 123
7.3 Solution of the Linear Equation......Page 125
7.4 Controllability Condition......Page 126
7.5 Examples......Page 128
7.6 Controllability: Free Final Time......Page 129
7.7 Navigation Problem......Page 130
8.1 Introduction......Page 134
8.2 General Problem with No States......Page 135
8.3.1 First Differential Condition......Page 136
8.3.2 Second Differential Condition......Page 138
8.3.4 Strong Variations......Page 139
8.4 Examples......Page 141
8.5 Free Final Time and Continuous Optimal Control......Page 142
8.6 Discontinuous Optimal Control......Page 144
8.7 Integral Constraint......Page 148
8.7.1 First Differential Condition......Page 149
8.7.2 Second Differential Conditions......Page 151
8.7.3 Strong Variations......Page 152
8.7.4 Navigation Problem......Page 153
8.8 Control Equality Constraint......Page 154
8.9 Control Inequality Constraint......Page 156
9.2 Preliminary Remarks......Page 160
9.3 First Differentia] Conditions......Page 162
9.3.1 No Final State Constraints......Page 164
9.3.2 With Final State Constraints......Page 165
9.4 Summary......Page 167
9.6 Example......Page 168
9.7 Acceleration Problem......Page 170
9.8 Navigation Problem......Page 171
9.9 Minimum Distance on a Sphere......Page 172
10.1 Introduction......Page 186
10.2 Weierstrass Condition......Page 187
10.3 Legendre-Clebsch Condition......Page 189
10.4 Examples......Page 190
11.2 The Second Differential......Page 193
11.3 Legendre-Clebsch Condition......Page 196
11.4 Neighboring Optimal Paths......Page 197
11.5 Neighboring Optimal Paths on a Sphere......Page 202
11.6 Second Differential Condition......Page 203
11.8 Acceleration Problem......Page 208
11.10 Minimum Distance on a Sphere......Page 210
11.11 Minimum Distance Between Two Points on a Sphere......Page 212
11.12 Other Sufficient Conditions......Page 214
12.1 Introduction......Page 219
12.2 Optimal Guidance......Page 220
12.3 Neighboring Optimal Guidance......Page 222
12.4 Transition Matrix Method......Page 224
12.4.2 Symplectic Property of the Transition Matrix......Page 225
12.4.3 Solution of the Linear TPBVP......Page 226
12.4.4 Relationship to the Sweep Method......Page 228
12.5 Linear Quadratic Guidance......Page 230
12.5.1 Sweep Solution......Page 231
12.5.2 Transition Matrix Solution......Page 232
12.6 Homing Missile Problem......Page 233
13.2 First Differential Conditions......Page 241
13.4 Second Differential......Page 244
13.5 Neighboring Optimal Paths......Page 246
13.6 Second Differential Conditions......Page 250
13.7 Example......Page 252
13.8 Distance Problem......Page 255
13.9 Navigation Problem......Page 258
14.2 Problem Formulation......Page 267
14.3 Controllability......Page 269
14.4 First Differential Conditions......Page 270
14.5 Second Differential Conditions......Page 271
14.6 Navigation Problem......Page 275
15.2 Problem Statement......Page 278
15.3 First Differential Conditions......Page 279
15.4 Tests for a Minimum......Page 280
15.5 Second Differential Conditions......Page 281
15.6 Minimum Distance Between a Parabola and a Line......Page 284
15.7 Parameters as States......Page 287
15.8 Navigation Problem......Page 288
15.9 Partitioning the Parameter Problem......Page 290
15.10 Navigation Problem......Page 292
16.1 Introduction......Page 295
16.2 Problem Statement......Page 296
16.3 First Differential Conditions......Page 297
16.4 Tests for a Minimum......Page 301
16.5 Example......Page 302
16.6 Second Differential......Page 303
16.7 Neighboring Optimal Path......Page 305
16.8 Second Differential Conditions......Page 307
16.9 Supersonic Airfoil of Minimum Pressure Drag......Page 308
17.1 Introduction......Page 313
17.3 Control Equality Constraint......Page 314
17.4 State Equality Constraint......Page 316
17.5 Control Inequality Constraint......Page 318
17.6 Example......Page 322
17.7 Acceleration Problem......Page 325
17.8 Alternate Approach for $\bar{C}(t,x,u) \leq 0$......Page 329
17.9 State Inequality Constraint......Page 330
17.10 Example......Page 332
Part III. Approximate Solutions......Page 337
18.1 Introduction......Page 338
18.2 Algebraic Perturbation Problem......Page 339
18.3 Expansion Process for the General Problem......Page 340
18.4 Differential Process for the General Problem......Page 341
18.6 Expansion Process for a Particular Problem......Page 342
18.7 Differential Process for a Particular Problem......Page 343
18.8 Another Example......Page 344
18.9 Remarks......Page 346
19.1 Introduction......Page 347
19.2 Regular Perturbation Problem......Page 348
19.3 Initial Value Problem with Fixed Final Time......Page 349
19.4 Initial Value Problem with Free Final Time......Page 352
19.5 Motion Relative to an Oblate Spheroid Earth......Page 354
19.6 Clohessy-Wiltshire Equations......Page 355
19.7 Remarks......Page 357
20.2 Optimal Control Problem with a Small Parameter......Page 358
20.3 Application to a Particular Problem......Page 360
20.4 Application to a General Problem......Page 364
20.5 Solution by the Sweep Method......Page 365
20.6 Navigation Problem......Page 367
20.7 Remarks......Page 369
21.1 Introduction......Page 370
21.2 Optimization Problems......Page 371
21.3 Explicit Numerical Integration......Page 373
21.4 Conversion with $a$, $u_k$ as Unknowns......Page 376
21.5 Conversion with $a$, $u_k$, $x_j$ as Unknowns......Page 377
21.6 Implicit Numerical Integration......Page 378
21.7 Conversion with $a$, $u_k$, $x_k$ as Unknowns......Page 382
21.8 Conversion with $a$, $x_k$ as Unknowns......Page 384
21.9 Remarks......Page 385
Appendix A First and Second Differentials by Taylor Series Expansion......Page 386
References......Page 393
Index......Page 397
Back cover......Page 402
