Hybrid Systems, Optimal Control and Hybrid Vehicles (eBook)

Dr. Thomas Böhme is currently working as a technical consultant at the iav automotive engineering company. He has over 15 years industrial experience in modeling and control of chemical processes and embedded systems. His research interests are vehicle control and optimal control of hybrid systems. He published over 30 peer-reviewed papers, journal contributions and patents. Benjamin Frank is mathematician and working at the iav automotive engineering company. He published several papers and contributions in the field of control and optimizations.

Series Editors’ Foreword 6
Preface 8
Intended Readership 8
What are the Contributions of This Book 9
What is Not Covered in This Book 10
Structure of the Book 10
Acknowledgements 14
Contents 15
Abbreviations and Symbols 22
1 Introduction 33
1.1 Motivation, Challenges, and Objectives 33
1.2 Vehicle Design Aspects 35
1.2.1 Stages of Energy Conversion 36
1.2.2 Real-World Driving Profile, Consumption, and Emissions 40
1.3 Process Model, Control Strategy, and Optimization 42
1.3.1 General Problem Statement 42
1.3.2 Energy Management 44
1.3.3 Numerical Solutions 48
1.4 Bibliographical Notes 51
References 52
Part I Theory and Formulations 56
2 Introduction to Nonlinear Programming 57
2.1 Introduction 57
2.2 Unconstrained Nonlinear Optimization 60
2.2.1 Necessary and Sufficient Conditions for Optimality 61
2.2.2 Newton--Raphson Method 61
2.2.3 Globalization of the Newton--Raphson Method 64
2.2.4 Quasi-Newton Method 67
2.3 Constrained Nonlinear Optimization 69
2.3.1 Necessary and Sufficient Conditions for Optimality 71
2.3.2 Projected Hessian 74
2.3.3 Sequential Quadratic Programming 76
2.4 Sensitivity Analysis 84
2.4.1 Sensitivity Analysis of the Objective Function and Constraints 88
2.4.2 Linear Perturbations 93
2.4.3 Approximation of the Perturbed Solution 94
2.4.4 Approximation of the Confidence Region 96
2.5 Multi-Objective Optimization 97
2.5.1 Elitist Multi-Objective Evolutionary Algorithm 98
2.5.2 Remarks for MOGAs 102
2.6 Bibliographical Notes 103
References 104
3 Hybrid Systems and Hybrid Optimal Control 108
3.1 Introduction 108
3.2 System Definition 109
3.2.1 Continuous Systems 109
3.2.2 Hybrid Systems 112
3.2.3 Controlled Hybrid Systems and Switched Systems 115
3.2.4 Existence and Uniqueness of Admissible States and Controls 117
3.2.5 Control and State Constraints, Admissible Sets, and Admissible Function Spaces 120
3.2.6 Reformulation of Switched Systems 123
3.3 Optimal Control Problem Formulations 125
3.3.1 Functionals 125
3.3.2 Boundary Conditions 126
3.3.3 Continuous Optimal Control Problem 127
3.3.4 Hybrid Optimal Control Problem 129
3.3.5 Switched Optimal Control Problem 130
3.3.6 Binary Switched Optimal Control Problem 131
3.3.7 Transformations of Optimal Control Problems 132
3.4 Bibliographical Notes 139
References 141
4 The Minimum Principle and Hamilton--Jacobi--Bellman Equation 145
4.1 Introduction 145
4.1.1 The Calculus of Variations 145
4.1.2 Deriving First-Order Necessary Conditions for an Extremum of an Optimal Control Problem 148
4.2 Minimum Principle 153
4.2.1 Necessary Conditions for Optimal Control Problems with Control Restraints 156
4.2.2 Necessary Conditions for Optimal Control Problems with State Constraints 159
4.2.3 Necessary Conditions for Optimal Control Problems with Affine Controls 165
4.3 Hamilton--Jacobi--Bellman Equation 168
4.4 Hybrid Minimum Principle 174
4.4.1 Necessary Conditions for Switched Optimal Control Problems Without State Jumps 179
4.4.2 Necessary Conditions for Switched Optimal Control Problems with State Jumps 180
4.4.3 Revisited: Necessary Conditions for a State Constrained Optimal Control Problem 181
4.5 Existence 184
4.6 Bibliography 187
References 189
Part II Methods for Optimal Control 192
5 Discretization and Integration Schemes for Hybrid Optimal Control Problems 193
5.1 Introduction 193
5.2 Discretization of the Initial Value Problem 194
5.3 Runge--Kutta Integration Scheme 195
5.4 Consistence Order of Runge--Kutta Methods 200
5.5 Stability 209
5.6 Some Lower-Order Runge--Kutta Integration Schemes 211
5.6.1 Explicit Runge--Kutta Schemes 212
5.6.2 Implicit Runge--Kutta Schemes 215
5.7 Remarks for Integration Schemes for Switched System with Discontinuities 220
5.8 Consequences of the Discretization to Optimal Control Problems 221
5.9 Bibliographical Notes 222
References 223
6 Dynamic Programming 225
6.1 Introduction 225
6.2 Optimal Control for Continuous Systems 226
6.3 Optimal Control of Hybrid Systems 232
6.4 Discussion 236
6.5 Bibliography 238
References 239
7 Indirect Methods for Optimal Control 241
7.1 Introduction 241
7.2 Optimal Control for Continuous Systems 242
7.2.1 Indirect Shooting Method 242
7.2.2 Indirect Multiple Shooting Method 247
7.3 Optimal Control for Hybrid Systems 251
7.4 Discussion 254
7.5 Bibliography 256
References 257
8 Direct Methods for Optimal Control 258
8.1 Introduction 258
8.2 Optimal Control for Continuous Systems 264
8.2.1 Direct Shooting 265
8.2.2 Direct Collocation 270
8.2.3 Comparison of Direct Shooting and Direct Collocation 272
8.2.4 Recovering the Costates from a Direct Shooting and Direct Collocation 272
8.3 Optimal Control for Switched Systems 274
8.3.1 Embedded Optimal Control Problem 275
8.3.2 Two-Stage Algorithm 278
8.3.3 Switching Time Optimization with Parameterized Switching Intervals 282
8.4 Numerical Methods for Obtaining Binary Feasible Control Functions 286
8.5 Discussion 291
8.6 Bibliography 292
References 295
Part III Numerical Implementations 299
9 Practical Implementation Aspects of Large-Scale Optimal Control Solvers 300
9.1 Sparse Linear Algebra 300
9.1.1 Sparse Matrix Formats 300
9.1.2 Numerical Solution of Large-Scale Linear Systems 301
9.1.3 Checking the Positive Definiteness of Large-Scale Matrices 305
9.2 Calculating Derivatives 306
9.2.1 Computational Graphs 306
9.2.2 Sparsity Pattern Determination 307
9.2.3 Compressed Derivative Calculation 311
9.2.4 Finite Differences 314
9.3 Sparse Quasi-Newton Updates 318
9.3.1 Quasi-Newton Update for Partially Separable Function 318
9.3.2 Simple Quasi-Newton Update for Chordal Sparsity Structures 319
9.3.3 Quasi-Newton Update for Chordal Sparsity Structures 321
9.3.4 Modifications of the Quasi-Newton Update 323
9.3.5 Quasi-Newton Updates for Discretized Optimal Control Problems 324
9.4 Bibliographical Notes 326
References 327
Part IV Modeling of Hybrid Vehicles for Control 330
10 Modeling Hybrid Vehicles as Switched Systems 331
10.1 Introduction 331
10.2 Vehicle Dynamics 333
10.3 Mechatronic Systems 336
10.3.1 Internal Combustion Engine 337
10.3.2 Electric Machine 342
10.3.3 Gearbox 349
10.3.4 Clutch 355
10.3.5 Battery 356
10.4 Hybrid Vehicle Configurations 363
10.4.1 Parallel Hybrids 365
10.4.2 Power-Split Hybrids 372
10.4.3 Serial Hybrids 388
10.4.4 Combined Hybrids 392
10.4.5 Plug-In Hybrids 393
10.4.6 Battery Electric Vehicles 394
10.5 Hybrid Vehicle Models 395
10.5.1 Quasi-static Model for Parallel Hybrids 396
10.5.2 Thermodynamic Model for Parallel Hybrids Using Spark Ignition Engines 399
10.5.3 Quasi-static Model for Power-Split Hybrids 404
10.5.4 Extended Quasi-static Model for Parallel Hybrids 407
10.6 Drive Cycles 407
10.7 Static Function Representation 413
10.8 Switching Costs 413
10.9 Bibliographical Notes 415
References 417
Part V Applications 421
11 Advanced Vehicle Calibration 422
11.1 Introduction 422
11.2 Offline Solution of Switched Optimal Control Problems ƒ 422
11.3 Analytical Calibration for Rule-Based Energy Managements 433
11.3.1 Constant Costate Assumption 435
11.3.2 Influence of Switching Costs 437
11.3.3 Lookup Table Calculation 438
11.4 Rule-Based Strategies for Choosing the Costate 442
11.4.1 Rule-Based Selection Using Costate Maps 443
11.4.2 Costate for Optimal CO2 Emissions 444
11.5 Implementation Issues 445
11.6 Bibliography 447
References 448
12 Predictive Real-Time Energy Management 450
12.1 Introduction 450
12.2 Real-World Benchmark-Cycles 452
12.3 Intelligent Traffic System 454
12.3.1 Time-Based Driver Model 455
12.3.2 Spatial-Based Driver Model 457
12.3.3 Estimation of Stop Events 460
12.4 Predictive Energy Management for Battery Electric Vehicles 461
12.4.1 Vehicle Model 463
12.4.2 Dynamic Programming for the Maximal Speed Limit 464
12.4.3 Instantaneous Speed Limit Corrections 466
12.4.4 Experimental Results 467
12.5 Predictive Energy Management for Hybrid Vehicles 468
12.5.1 Event-Triggered Predictive Energy Management 471
12.5.2 Predictive Energy Management with Long Prediction Horizon 481
12.6 Bibliographical Notes 495
References 499
13 Optimal Design of Hybrid Powertrain Configurations 502
13.1 Introduction 502
13.2 Process Description 503
13.2.1 Drivability Performance Index 503
13.2.2 Design Parameters 504
13.2.3 Powertrain Dynamics 505
13.3 Multi-objective Powertrain Design 507
13.3.1 Master Problem 509
13.3.2 Map Scaling for Powertrain Components 509
13.3.3 Batched Optimal Control Subproblems 512
13.4 P2-Hybrid Design Study 519
13.5 Post Optimal Parametric Sensitivity Analysis 528
13.6 Further Work 534
13.6.1 Speedup of the Algorithm 534
13.6.2 Increase of Model Complexity 536
13.7 Bibliographical Notes 536
References 537
Part VI Appendix 540
14 Graph Theoretical Fundamentals for Sparse Matrices 541
References 545
Index 546