Nonlinear Phenomena in Power Electronics:attractors,bifurcations,chaos,and nonlinear control

1.1 Introduction to Power Electronics......Page 16
Index......Page 0
1.1.1 Power Switching Devices......Page 17
1.1.2.2 Electrical Machines and Drives......Page 19
1.2 An Example: The Buck DC/DC Converter......Page 20
1.2.1 Conventional Model of the Buck Converter......Page 21
1.2.1.3 Perturbation......Page 22
1.2.1.7 Stability......Page 23
1.2.3 Nonlinear Map-Based Model of the Buck Converter......Page 24
1.2.4 Discontinuous Conduction Mode......Page 28
1.2.5 Limitations and Extensions of Average Models......Page 29
1.3 Study of Nonlinear Dynamics and Chaos in Power Electronics......Page 30
References......Page 35
2.1.2.1 High-Frequency PWM DC/DC Converters......Page 40
2.1.2.2 Other High-Frequency PWM Converters......Page 41
2.1.2.4 Resonant Converters......Page 42
2.1.3 Averaged and Sampled-Data Models for Analysis, Simulation, and Control of Converter Dynamics......Page 43
2.1.3.1 Switched State-Space Model for the Boost Converter......Page 45
2.1.3.2 Averaged Model for the Boost Converter......Page 47
2.1.3.3 Averaged Model for Current-Mode Control of the Boost Converter......Page 48
2.1.3.4 Sampled-Data Models for the Boost Converter......Page 49
2.1.4.2 Generalized State-Space Models......Page 51
References......Page 52
2.2.2 Poincaré Maps for Smooth and Nonsmooth Dynamical Systems......Page 53
2.2.3 Piecewise-Smooth Power Electronic Circuits......Page 55
2.2.4 Power Electronic Systems as Hybrid Systems......Page 58
2.2.5 Stroboscopic Maps......Page 61
2.2.5.3 Closed-Loop Maps......Page 63
2.2.6.1 S-Switching Maps for DC/DC Converters......Page 64
2.2.6.2 A-Switching Maps for DC/DC Converters......Page 65
2.2.8 Conclusions......Page 66
References......Page 67
3.1.1 System State, and State-Space Models......Page 68
3.1.3 Vector Fields of Linear, Linearized, and Nonlinear Systems......Page 70
3.1.4 Attractors in Nonlinear Systems......Page 72
3.1.5 Chaos......Page 74
3.1.6 Poincaré Map......Page 76
3.1.7 Dynamics of Discrete-Time Systems......Page 77
3.1.8 Fractal Geometry......Page 79
3.1.10 Bifurcation......Page 81
3.2 Bifurcations of Smooth Maps......Page 82
3.2.2 The Saddle-Node Bifurcation......Page 83
3.2.3 The Period-Doubling Bifurcation......Page 84
3.2.4 The Neimark Bifurcation......Page 85
3.3 Bifurcations in Piecewise-Smooth Maps......Page 88
3.3.1 The Normal Form......Page 92
3.3.2 Bifurcations in the One-Dimensional Normal Form......Page 94
3.3.2.1 Border Collision Pair Bifurcation......Page 95
3.3.2.2 Border-Crossing Bifurcations......Page 96
3.3.3 Bifurcations in the Two-Dimensional Normal Form......Page 97
3.3.4.1 Border Collision Pair Bifurcation......Page 99
3.3.4.2 Border-Crossing Bifurcations......Page 100
References......Page 103
3.4 Nonstandard Bifurcations in Discontinuous Maps......Page 104
3.5.2 Problem Description......Page 109
3.5.3 Mechanism of Period Doubling......Page 110
3.5.4 Schwarzian Derivative and Period Doublings......Page 111
3.5.5 Application to Power Electronics......Page 112
3.5.6 Illustrative Example: The Boost Converter......Page 113
References......Page 115
3.6.1 Characteristic (Floquet) Multipliers......Page 116
3.6.2 Invariant Sets and Invariant Manifolds......Page 117
3.6.2.1 Homoclinic and Heteroclinic Orbits......Page 118
3.6.3 Coexisting Attractors......Page 119
3.6.4 The Role of Invariant Manifolds and Basins of Attraction......Page 120
3.6.5 Crises......Page 121
3.6.5.1 Interior Crises......Page 122
3.6.5.2 Boundary Crises......Page 123
References......Page 124
4.1.2 Overview of Simulation Study and Verification......Page 126
4.1.4 Displaying Time-Domain Waveforms, Attractors, and Spectra......Page 127
4.1.5 Displaying Poincaré Sections......Page 129
4.1.5.1 Principle of Poincaré Section Measurement......Page 130
4.1.5.2 Example: Free-Running Cuk Converter......Page 132
4.1.7 Displaying Bifurcation Diagrams......Page 133
4.1.7.1 Basic Operational Requirements......Page 134
4.1.7.2 Digital Implementation and Related Issues......Page 135
4.1.7.4 Example: Boost Converter under Current-Mode Control......Page 136
4.2.1 Simulation of Power Electronic Circuits......Page 138
4.2.1.1 Problems Arising from Varying Topology......Page 139
4.2.2 Obtaining Bifurcation Diagrams......Page 140
4.2.3 Plotting Basins of Attraction in Systems with Multiple Attractors......Page 141
4.2.4 Computing the Maximal Lyapunov Exponent......Page 142
4.3.1 Introduction......Page 144
4.3.2.1 Describing Chaotic Behavior via Densities......Page 145
4.3.2.2 Calculation of the Time-Average of the Inductor Current......Page 149
4.3.2.3.1 Boost Converter......Page 150
4.3.2.3.2 Buck-Boost Converter......Page 153
4.3.2.3.3 Buck Converter......Page 154
4.3.2.3.4 Cuk Converter......Page 156
4.3.2.3.5 Average Switching Frequency and Average Duty Ratio......Page 157
4.3.3 Experimental Results......Page 158
Appendix: The Frobenius-Perron Operator......Page 160
References......Page 163
4.4.1 Characterization of Spectral Properties......Page 164
53831_04b......Page 165
4.4.2 Motivation and Outline......Page 166
4.4.3 The Simplified Mapping......Page 167
4.4.4 Approximation of the Mean State Variables......Page 169
4.4.5 The Power Density Spectrum of the Inductor Current......Page 170
4.4.6 The Invariant Density Algorithm......Page 173
4.4.8 Experimental Results......Page 175
4.4.9 Discussion......Page 178
References......Page 179
4.5.2 Nonlinear Systems and Stability of Periodic Solutions......Page 180
4.5.3 Computer Methods to Analyze Stability......Page 182
4.5.4 Computation of the Jacobian Matrix......Page 184
4.5.5 Analysis Method Based on Transient Simulator......Page 186
4.5.6.1 Classification of Bifurcations......Page 189
4.5.6.2 Method to Determine Bifurcation Values......Page 190
References......Page 191
4.6.1 Introduction......Page 192
4.6.2 What is Operating-Mode Analysis?......Page 193
4.6.3.1 Conditions that Define Operating-Mode Boundaries......Page 194
4.6.3.2 Numerical Tracing of Boundary Curves......Page 196
4.6.3.3 Computation of Steady-State Sensitivities......Page 197
4.6.3.4 Numerical Examples......Page 198
4.6.4.1 Basic Approach......Page 202
4.6.4.3 Application to More Complicated Converters......Page 203
References......Page 205
5.1.1 Modeling and Analysis......Page 207
5.1.2 Analysis of Bifurcations......Page 210
References......Page 213
5.2.1 Overview of Circuit Operation......Page 214
5.2.2 Experimental Results......Page 215
5.2.3 Coexisting Attractors and Crises......Page 217
References......Page 222
5.3.1 Buck Converter Modeling under Voltage-Mode Control......Page 223
5.3.1.1 Differential Equations......Page 224
5.3.2 Discrete-Time Map and Periodic Orbits......Page 225
5.3.2.2 Analytical Study of Periodic Orbits: Existence and Stability......Page 227
5.3.3.1 The Main Bifurcation Diagram......Page 232
5.3.3.2 Secondary Bifurcations......Page 234
5.3.4.2 Invariant Manifolds and Basins of Attraction......Page 235
5.3.4.3 6T-Periodic Orbits......Page 238
5.3.4.4 12T-Periodic Orbits......Page 239
5.3.4.5 5T-Periodic Orbits and the Jump to Larger Chaos......Page 240
References......Page 243
5.4.2 General Sampled-Data Model for Closed-Loop PWM Converters......Page 244
5.4.4 Saddle-Node Bifurcation in Buck Converter under Discrete-Time Control......Page 247
5.4.5 Neimark Bifurcation in Buck Converter under Voltage-Mode Control......Page 249
5.4.6 Neimark Bifurcation in Buck Converter with Input Filter under Voltage-Mode Control......Page 251
5.4.7 Neimark Bifurcation in Buck Converter with Input Filter under Current-Mode Control......Page 253
5.5.1 Review of Operating Modes......Page 255
5.5.2 Derivation of Discrete-Time Maps......Page 256
5.5.3 Period-Doubling Bifurcation......Page 258
5.5.4 Computer Simulations and Experiments......Page 260
5.5.5 Remarks and Summary......Page 261
5.6.1 Review of the Cuk Converter and its Operation......Page 263
5.6.2 Discrete-Time Modeling for Fixed Frequency Operation......Page 264
5.6.3.1 Autonomous System Modeling......Page 266
5.6.3.2 Dimensionless Equations......Page 268
5.6.3.3 Stability of Equilibrium Point and Hopf Bifurcation......Page 269
5.6.3.4 Local Trajectories from Describing Equation......Page 271
5.6.3.5 Computer Simulation Study......Page 273
References......Page 276
6.1 Introduction......Page 277
6.3 Static VAR System Example......Page 278
6.4 Poincaré Map......Page 280
6.5 Jacobian of Poincaré Map......Page 282
6.5.1 Thyristor Current Function and Transversality......Page 283
6.5.2 Relations between On and Off Systems......Page 284
6.5.3.1 Interval Containing a Switch-on......Page 285
6.5.3.2 Interval Containing a Switch-off......Page 286
6.5.3.3 Assembling the Jacobian......Page 287
6.5.4 Discussion of Jacobian Formula......Page 288
6.6.1 Simple Example......Page 289
6.6.2 Switching Damping in the SVC Example......Page 290
6.6.3 Variational Equation......Page 291
6.7.1 Switching Time Bifurcations and Instability......Page 293
6.7.2 Switching Time Bifurcations for Transients......Page 295
6.7.3 Misfire Onset as a Transcritical Bifurcation......Page 296
6.7.4 Noninvertibility and Discontinuity of the Poincaré Map......Page 298
6.7.5 Multiple Attractors and Their Basin Boundaries......Page 299
6.8.2 Poincaré Map Jacobian for the DC/DC Buck-Boost Converter in Discontinuous Mode......Page 301
6.8.3 Poincaré Map Continuity and Switching Time Bifurcations......Page 303
6.9 Firing Angle Control......Page 304
References......Page 305
7.1.2 The Circuit......Page 307
7.1.3 Saturating and Hysteretic Inductor Modeling......Page 308
7.1.4 Differential Equation for the Circuit......Page 309
7.1.5 Results......Page 310
7.1.5.1 Bifurcation Diagram Comparison......Page 311
7.1.5.2 Poincaré Section Comparison......Page 312
7.2.1 Introduction......Page 313
7.2.2 Functioning Principle......Page 314
7.2.3 System Model and Equation......Page 315
7.2.4 Poincaré Map......Page 317
7.2.6 Mode of Oscillations, Bifurcations, and Crises......Page 318
7.2.6.1 Chaotic Mode......Page 319
7.2.6.2 Symmetry-Breaking Bifurcation......Page 320
7.2.6.3 Merging Crisis......Page 322
7.2.6.5 Saddle-Node Bifurcation and Square-Wave Mode......Page 323
7.2.6.6 Boundary Crisis......Page 326
References......Page 327
7.3.1 Introduction......Page 328
7.3.2 Discussion of Switching Noise......Page 329
7.3.3.1 External Noise Action......Page 330
7.3.3.2 Background of Analysis......Page 331
7.3.3.4 Probability Density Function of Switch Timing......Page 333
7.3.3.5 Implications of the Non-Gaussian Duty Ratio Distribution with Latch......Page 334
7.3.4.2 The Closed-Loop Process and System Model......Page 336
7.3.4.3 Time Domain Noise Analysis......Page 337
7.3.4.4 Confirmation......Page 338
7.3.4.5 Frequency Domain Analysis......Page 339
7.3.5 Summary......Page 342
7.4 Nonlinear Phenomena in the Current Control of Induction Motors......Page 343
7.4.1.1 Voltage Source Converter (VSC)......Page 344
7.4.1.2 AC Side......Page 345
7.4.1.3 Hysteresis Current Control (HCC)......Page 346
7.4.1.4 Poincaré Map......Page 347
7.4.2.2 Period Doubling Bifurcation......Page 348
7.4.2.3 Intermittency......Page 350
7.4.3 Numerical Values......Page 351
References......Page 352
7.5.2.1 Model of Induction Motor and its Mechanical Load......Page 353
7.5.2.2 Model of Inverter and Rectifier......Page 355
7.5.3.1 Case I......Page 358
7.5.3.2 Case II......Page 359
7.5.4 Stability Analysis......Page 360
7.5.5 Analysis of Bifurcations......Page 363
References......Page 367
8.1.1 Introduction......Page 368
8.1.3 Nonlinear Modulation......Page 369
References......Page 371
8.2.1 Introduction......Page 372
8.2.2 Hysteresis Control......Page 373
8.2.3.1 Trajectories and Equilibria......Page 375
8.2.3.2 Switching Surface-Based Control Laws......Page 376
8.2.3.3 Necessary Conditions for Switching Surface Controls......Page 377
8.2.3.4 Sample Outputs and Hysteresis Design Approaches......Page 378
8.2.4.1 Successor Points......Page 380
8.2.4.2 Behavior near a Switching Surface......Page 381
8.2.4.3 Choosing a Switching Surface......Page 382
References......Page 384
8.3.1 Introduction......Page 386
8.3.2.1 Basic Control......Page 387
8.3.2.2 Adaptation......Page 389
8.3.2.3 Estimation and Output Feedback......Page 390
8.3.3.1 Basic Controller......Page 391
8.3.3.2 Adaptation......Page 393
8.3.3.3 Hamiltonian Control......Page 394
8.3.4 Connections with Sliding-Mode Control......Page 395
8.3.4.1 Sliding-Mode Controller Revisited......Page 396
8.3.4.2 Passivity-Based Sliding-Mode Controller......Page 397
8.3.4.3 Combining SMC with Prediction......Page 398
8.3.5 Conclusions......Page 399
References......Page 400
8.4.2 Ripple-Based Control......Page 401
8.4.3 Ripple Correlation......Page 402
8.4.4.1 Adaptive Dead Time......Page 404
8.4.4.2 Solar Power Processing......Page 405
8.4.4.3 Motor Power Minimization in Drives......Page 406
References......Page 407
8.5.1 Introduction......Page 408
8.5.2 A Combination of OGY and Pyragas Methods......Page 409
8.5.2.1 Application to the Current-Mode-Controlled Boost Converter......Page 410
8.5.3 Controlling Border-Collision Bifurcations......Page 413
8.5.3.1 Local Feedback Strategy......Page 414
8.5.3.2 An Example: A Two-Dimensional Map......Page 415
8.5.4 Time-Delay Control of Chaos......Page 416
8.5.4.1 An Example: TDAS for the Current-Mode Boost Converter......Page 417
References......Page 419
8.6.2 Dynamics and Control......Page 421
8.6.3 Experimental Results......Page 423
8.6.4 The OGY Method......Page 424
8.6.4.1 Review of the OGY Method......Page 426
8.6.4.2 Controlling DC/DC Converters......Page 427
8.6.4.3 Integral Control......Page 430
Acknowledgment......Page 432
8.7.1 Background......Page 433
8.7.2 Controlling Bifurcation in Discontinuous-Mode Converters......Page 434
8.7.3.1 Use of Compensating Ramp for Controlling Bifurcation......Page 435
8.7.3.2 Effects on Dynamical Response......Page 439
8.7.3.3 Experimental Measurements......Page 440
Acknowledgment......Page 441
8.8.1 Background......Page 443
8.8.2 The Drive-Response Concept......Page 444
8.8.3 Synchronization in Chaotic Free-Running Cuk Converters......Page 445
8.8.4 Derivation of the Conditional Lyapunov Exponents......Page 446
8.8.5 Numerical Calculation of the Conditional Lyapunov Exponents......Page 447
8.8.7 Remarks on Practical Synchronization......Page 448
References......Page 450
B......Page 452
D......Page 454
E......Page 455
G......Page 456
I......Page 457
M......Page 458
P......Page 459
S......Page 460
U......Page 462
Z......Page 463
About the Editors......Page 464
Front Matter......Page 1
Table of Contents......Page 3
Preface......Page 13