Optimal control of hybrid vehicles

Series Editors’ Foreword......Page 6
 Intended Readership......Page 8
 What are the Contributions of This Book......Page 9
 Structure of the Book......Page 10
Acknowledgements......Page 14
Contents......Page 15
Abbreviations and Symbols......Page 22
 1.1 Motivation, Challenges, and Objectives......Page 33
 1.2 Vehicle Design Aspects......Page 35
  1.2.1 Stages of Energy Conversion......Page 36
  1.2.2 Real-World Driving Profile, Consumption,  and Emissions......Page 40
  1.3.1 General Problem Statement......Page 42
  1.3.2 Energy Management......Page 44
  1.3.3 Numerical Solutions......Page 48
 1.4 Bibliographical Notes......Page 51
 References......Page 52
Part I Theory and Formulations......Page 56
 2.1 Introduction......Page 57
 2.2 Unconstrained Nonlinear Optimization......Page 60
  2.2.2 Newton--Raphson Method......Page 61
  2.2.3 Globalization of the Newton--Raphson Method......Page 64
  2.2.4 Quasi-Newton Method......Page 67
 2.3 Constrained Nonlinear Optimization......Page 69
  2.3.1 Necessary and Sufficient Conditions for Optimality......Page 71
  2.3.2 Projected Hessian......Page 74
  2.3.3 Sequential Quadratic Programming......Page 76
 2.4 Sensitivity Analysis......Page 84
  2.4.1 Sensitivity Analysis of the Objective Function  and Constraints......Page 88
  2.4.2 Linear Perturbations......Page 93
  2.4.3 Approximation of the Perturbed Solution......Page 94
  2.4.4 Approximation of the Confidence Region......Page 96
 2.5 Multi-Objective Optimization......Page 97
  2.5.1 Elitist Multi-Objective Evolutionary Algorithm......Page 98
  2.5.2 Remarks for MOGAs......Page 102
 2.6 Bibliographical Notes......Page 103
 References......Page 104
 3.1 Introduction......Page 108
  3.2.1 Continuous Systems......Page 109
  3.2.2 Hybrid Systems......Page 112
  3.2.3 Controlled Hybrid Systems and Switched Systems......Page 115
  3.2.4 Existence and Uniqueness of Admissible States and Controls......Page 117
  3.2.5 Control and State Constraints, Admissible Sets,  and Admissible Function Spaces......Page 120
  3.2.6 Reformulation of Switched Systems......Page 123
  3.3.1 Functionals......Page 125
  3.3.2 Boundary Conditions......Page 126
  3.3.3 Continuous Optimal Control Problem......Page 127
  3.3.4 Hybrid Optimal Control Problem......Page 129
  3.3.5 Switched Optimal Control Problem......Page 130
  3.3.6 Binary Switched Optimal Control Problem......Page 131
  3.3.7 Transformations of Optimal Control Problems......Page 132
 3.4 Bibliographical Notes......Page 139
 References......Page 141
  4.1.1 The Calculus of Variations......Page 145
  4.1.2 Deriving First-Order Necessary Conditions for an Extremum of an Optimal Control Problem......Page 148
 4.2 Minimum Principle......Page 153
  4.2.1 Necessary Conditions for Optimal Control Problems with Control Restraints......Page 156
  4.2.2 Necessary Conditions for Optimal Control Problems with State Constraints......Page 159
  4.2.3 Necessary Conditions for Optimal Control Problems with Affine Controls......Page 165
 4.3 Hamilton--Jacobi--Bellman Equation......Page 168
 4.4 Hybrid Minimum Principle......Page 174
  4.4.1 Necessary Conditions for Switched Optimal Control Problems Without State Jumps......Page 179
  4.4.2 Necessary Conditions for Switched Optimal Control Problems with State Jumps......Page 180
  4.4.3 Revisited: Necessary Conditions for a State Constrained Optimal Control Problem......Page 181
 4.5 Existence......Page 184
 4.6 Bibliography......Page 187
 References......Page 189
Part II Methods for Optimal Control......Page 192
 5.1 Introduction......Page 193
 5.2 Discretization of the Initial Value Problem......Page 194
 5.3 Runge--Kutta Integration Scheme......Page 195
 5.4 Consistence Order of Runge--Kutta Methods......Page 200
 5.5 Stability......Page 209
 5.6 Some Lower-Order Runge--Kutta Integration Schemes......Page 211
  5.6.1 Explicit Runge--Kutta Schemes......Page 212
  5.6.2 Implicit Runge--Kutta Schemes......Page 215
 5.7 Remarks for Integration Schemes for Switched System with Discontinuities......Page 220
 5.8 Consequences of the Discretization to Optimal Control Problems......Page 221
 5.9 Bibliographical Notes......Page 222
 References......Page 223
 6.1 Introduction......Page 225
 6.2 Optimal Control for Continuous Systems......Page 226
 6.3 Optimal Control of Hybrid Systems......Page 232
 6.4 Discussion......Page 236
 6.5 Bibliography......Page 238
 References......Page 239
 7.1 Introduction......Page 241
  7.2.1 Indirect Shooting Method......Page 242
  7.2.2 Indirect Multiple Shooting Method......Page 247
 7.3 Optimal Control for Hybrid Systems......Page 251
 7.4 Discussion......Page 254
 7.5 Bibliography......Page 256
 References......Page 257
 8.1 Introduction......Page 258
 8.2 Optimal Control for Continuous Systems......Page 264
  8.2.1 Direct Shooting......Page 265
  8.2.2 Direct Collocation......Page 270
  8.2.4 Recovering the Costates from a Direct Shooting  and Direct Collocation......Page 272
 8.3 Optimal Control for Switched Systems......Page 274
  8.3.1 Embedded Optimal Control Problem......Page 275
  8.3.2 Two-Stage Algorithm......Page 278
  8.3.3 Switching Time Optimization with Parameterized Switching Intervals......Page 282
 8.4 Numerical Methods for Obtaining Binary Feasible Control Functions......Page 286
 8.5 Discussion......Page 291
 8.6 Bibliography......Page 292
 References......Page 295
Part III Numerical Implementations......Page 299
  9.1.1 Sparse Matrix Formats......Page 300
  9.1.2 Numerical Solution of Large-Scale Linear Systems......Page 301
  9.1.3 Checking the Positive Definiteness of Large-Scale Matrices......Page 305
  9.2.1 Computational Graphs......Page 306
  9.2.2 Sparsity Pattern Determination......Page 307
  9.2.3 Compressed Derivative Calculation......Page 311
  9.2.4 Finite Differences......Page 314
  9.3.1 Quasi-Newton Update for Partially Separable Function......Page 318
  9.3.2 Simple Quasi-Newton Update for Chordal Sparsity Structures......Page 319
  9.3.3 Quasi-Newton Update for Chordal Sparsity Structures......Page 321
  9.3.4 Modifications of the Quasi-Newton Update......Page 323
  9.3.5 Quasi-Newton Updates for Discretized Optimal Control Problems......Page 324
 9.4 Bibliographical Notes......Page 326
 References......Page 327
Part IV Modeling of Hybrid Vehicles  for Control......Page 330
 10.1 Introduction......Page 331
 10.2 Vehicle Dynamics......Page 333
 10.3 Mechatronic Systems......Page 336
  10.3.1 Internal Combustion Engine......Page 337
  10.3.2 Electric Machine......Page 342
  10.3.3 Gearbox......Page 349
  10.3.4 Clutch......Page 355
  10.3.5 Battery......Page 356
 10.4 Hybrid Vehicle Configurations......Page 363
  10.4.1 Parallel Hybrids......Page 365
  10.4.2 Power-Split Hybrids......Page 372
  10.4.3 Serial Hybrids......Page 388
  10.4.4 Combined Hybrids......Page 392
  10.4.5 Plug-In Hybrids......Page 393
  10.4.6 Battery Electric Vehicles......Page 394
 10.5 Hybrid Vehicle Models......Page 395
  10.5.1 Quasi-static Model for Parallel Hybrids......Page 396
  10.5.2 Thermodynamic Model for Parallel Hybrids Using Spark Ignition Engines......Page 399
  10.5.3 Quasi-static Model for Power-Split Hybrids......Page 404
 10.6 Drive Cycles......Page 407
 10.8 Switching Costs......Page 413
 10.9 Bibliographical Notes......Page 415
 References......Page 417
Part V Applications......Page 421
 11.2 Offline Solution of Switched Optimal Control Problems ƒ......Page 422
 11.3 Analytical Calibration for Rule-Based Energy Managements......Page 433
  11.3.1 Constant Costate Assumption......Page 435
  11.3.2 Influence of Switching Costs......Page 437
  11.3.3 Lookup Table Calculation......Page 438
 11.4 Rule-Based Strategies for Choosing the Costate......Page 442
  11.4.1 Rule-Based Selection Using Costate Maps......Page 443
  11.4.2 Costate for Optimal CO2 Emissions......Page 444
 11.5 Implementation Issues......Page 445
 11.6 Bibliography......Page 447
 References......Page 448
 12.1 Introduction......Page 450
 12.2 Real-World Benchmark-Cycles......Page 452
 12.3 Intelligent Traffic System......Page 454
  12.3.1 Time-Based Driver Model......Page 455
  12.3.2 Spatial-Based Driver Model......Page 457
  12.3.3 Estimation of Stop Events......Page 460
 12.4 Predictive Energy Management for Battery Electric Vehicles......Page 461
  12.4.1 Vehicle Model......Page 463
  12.4.2 Dynamic Programming for the Maximal Speed Limit......Page 464
  12.4.3 Instantaneous Speed Limit Corrections......Page 466
  12.4.4 Experimental Results......Page 467
 12.5 Predictive Energy Management for Hybrid Vehicles......Page 468
  12.5.1 Event-Triggered Predictive Energy Management......Page 471
  12.5.2 Predictive Energy Management with Long Prediction Horizon......Page 481
 12.6 Bibliographical Notes......Page 495
 References......Page 499
 13.1 Introduction......Page 502
  13.2.1 Drivability Performance Index......Page 503
  13.2.2 Design Parameters......Page 504
  13.2.3 Powertrain Dynamics......Page 505
 13.3 Multi-objective Powertrain Design......Page 507
  13.3.2 Map Scaling for Powertrain Components......Page 509
  13.3.3 Batched Optimal Control Subproblems......Page 512
 13.4 P2-Hybrid Design Study......Page 519
 13.5 Post Optimal Parametric Sensitivity Analysis......Page 528
  13.6.1 Speedup of the Algorithm......Page 534
 13.7 Bibliographical Notes......Page 536
 References......Page 537
Part VI Appendix......Page 540
14 Graph Theoretical Fundamentals for Sparse Matrices......Page 541
 References......Page 545
Index......Page 546