Optimal Control Theory for Applications (Mechanical Engineering Series)

Series Preface......Page 5
Preface......Page 7
Contents......Page 11
1.2 Classification of Systems......Page 21
1.3 Parameter Optimization......Page 22
1.4 Optimal Control Theory......Page 24
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
2.4 Distance Problem......Page 49
2.5 Function of Two Independent Variables......Page 50
2.6 Examples......Page 55
2. 7 Historical Example......Page 56
2.8 Function of n Independent Variables......Page 58
Problems......Page 59
3.1 Introduction......Page 62
3.2 Function of Two Constrained Variables......Page 63
3.3 Distance Pr-oblem......Page 70
3.4 Function of n Constrained Variables......Page 72
3.5 Example......Page 73
Problems......Page 75
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
Problems......Page 92
5.2 Matrix Algebra......Page 96
5.3 Matrix Calculus......Page 98
5.5 Function of n Constrained Variables......Page 102
Problems......Page 104
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 6 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
Problems......Page 132
8.1 Introduction......Page 134
8.2 General Problem with No States......Page 135
8.3 Fixed Final Time and Continuous Optimal Control......Page 136
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
8.8 Control Equality Constraint......Page 154
8.9 Control Inequality Constraint......Page 156
Problems......Page 158
9.2 Preliminary Remarks......Page 160
9.3 First Differential Conditions......Page 162
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
Problems......Page 175
10.1 Introduction......Page 186
10.2. Weierstrass Condition......Page 187
10.3 Legendre-Clebsch Condition......Page 189
10.4 Examples......Page 190
Problems......Page 192
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
Problems......Page 215
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
12.5 Linear Quadratic Guidance......Page 230
12.6 Homing Missile Problem......Page 233
Problems......Page 238
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
Problems......Page 263
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
Problems......Page 277
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
Problems......Page 293
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
Problems......Page 311
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 C(t, x, u) <= 0......Page 329
17.9 State Inequality Constraint......Page 330
17.10 Example......Page 332
Problems......Page 334
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 Pro·blem......Page 360
20.4 Application to a General Problem......Page 364
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, uk as Unknowns......Page 376
21.5 Conversion with a, uk, Xj as Unknowns......Page 377
21.6 Implicit Numerical Integration......Page 378
21.7 Conversion with a, Uk, Xk as Unknowns......Page 382
21.8 Conversion with a, Xk 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