The Koopman operator in systems and control

Preface......Page 7
Contents......Page 13
Part I Concepts......Page 24
 1.1 Introduction......Page 25
 1.2 The Koopman Operator......Page 26
  1.2.1 Koopman Operator for Autonomous Discrete-Time and Continuous-Time Systems......Page 27
  1.2.2 Some Properties of the Koopman Operator......Page 29
  1.3.1 Koopman Eigenvalues and Eigenfunctions......Page 32
  1.3.2 Relationship with Geometric Properties......Page 35
  1.3.3 Koopman Modes......Page 37
  1.4.1 Finite-Dimensional Approximation of the Koopman Operator......Page 40
  1.4.2 Data-Driven Methods......Page 44
 1.5 Koopman Operator for Systems with Inputs......Page 47
  1.5.1 Discrete-Time Systems......Page 48
  1.5.2 Continuous-Time Systems......Page 49
 1.6 Perspectives in Control Theory......Page 51
 References......Page 52
 2.1 Introduction......Page 56
  2.2.1 Preliminaries......Page 57
  2.2.2 A Stability Property of the Koopman Operator......Page 59
  2.3.1 Spectrum of the Koopman Operator......Page 60
  2.3.2 General Results......Page 61
  2.3.3 Specific Cases......Page 63
  2.4.1 Finite-Dimensional Approximation of the Eigenfunctions......Page 66
  2.4.2 Polynomial Approximations Using Sum-of-Squares Programming......Page 68
  2.5.1 Lyapunov Functions Based on Koopman Eigenfunctions......Page 72
  2.5.2 General Lyapunov Functions and Approximation Errors......Page 74
 2.6 Conclusion......Page 77
 References......Page 78
 3.1 Introduction......Page 80
 3.2 Setup and Problem Definition......Page 81
  3.2.1 Estimation Problem......Page 83
  3.3.1 Koopman-Based Transformation......Page 84
  3.3.2 Bilinear Input–Output Koopman Observer Form......Page 85
 3.4 KOF-Based Observability Criterion......Page 87
  3.5.1 State Estimation Without Constraints......Page 88
  3.5.2 Constrained State Estimation......Page 89
  3.6.1 Discretized NCSE......Page 90
  3.6.2 Discretized KCSE......Page 91
  3.6.3 Solution Approaches......Page 93
  3.7.1 Example I......Page 95
  3.7.2 Example II......Page 96
 3.8 Conclusions......Page 98
 References......Page 99
 4.1 Toward Bilinear Form: Infinitesimal Koopman Operator on Function Space......Page 101
  4.1.1 Koopman Overview and Motivation......Page 102
  4.1.2 Koopman Canonical Transform: A Review......Page 103
 4.2 Bilinearizability of the Koopman Canonical Transform: Sufficient Conditions......Page 105
 4.3 Reachability Analysis of the Koopman Bilinear Form......Page 107
  4.3.1 Conditions for Reachability......Page 109
  4.4.1 Completely Bilinearizable System......Page 114
  4.4.2 Approximately Bilinearized System......Page 116
  4.4.3 Controllability of the System......Page 117
 References......Page 118
 5.1 Introduction......Page 119
 5.2 Cascaded Systems......Page 121
  5.2.1 Solution Operators......Page 122
  5.2.2 Asymptotic Equivalence......Page 123
  5.3.1 Principal Eigenfunctions of Component Systems......Page 125
  5.3.2 Principal Eigenfunctions for Cascaded Systems......Page 126
 5.4 Main Results......Page 128
  5.4.1 Theorems for Linear, Chained Cascades......Page 129
  5.4.2 Examples......Page 132
  5.4.3 Numerical Example......Page 134
  5.4.4 Theorems for Nonlinear, Chained Cascades......Page 135
 5.5 Conclusions......Page 137
  5.6.1 Proof of Theorem5.2: 0 Asymptotic Relative Error—Linear, Chained Cascades......Page 138
  5.6.2 Proof of Theorem5.3: Perturbation of Principal Eigenfunctions—Nominal, Linear System......Page 145
  5.6.4 Proof of Theorem5.5: Perturbation of Principal Eigenfunctions—Nominal, Nonlinear Cascades......Page 147
 References......Page 148
 6.1 Introduction......Page 150
  6.2.1 Skew Product Flow Formulation......Page 152
  6.2.2 Process Formulation......Page 155
 6.3 Stochastic Koopman Operator......Page 157
 6.4 Data-Driven Algorithms for the Nonautonomous Koopman Operator......Page 163
  6.4.1 Numerical Examples......Page 169
 6.5 Data-Driven Algorithms for the Stochastic Koopman Operator......Page 171
  6.5.1 Numerical Examples......Page 174
 References......Page 177
 7.1 Introduction......Page 180
  7.1.1 DMD Framework and Applications......Page 181
  7.2.1 Krylov Decomposition......Page 183
  7.2.2 The Curse of Ill-Conditioning......Page 186
  7.3.1 Low-Rank Approximation of the Data Matrix......Page 189
  7.3.2 DMD Algorithm......Page 190
  7.3.3 Numerical Computation of the SVD......Page 192
 7.4 An Enhanced DMD—Refined Rayleigh–Ritz DDMD with Residual Bound (DDMD_R4)......Page 195
  7.4.1 Data-Driven Residual and Refined Ritz Pairs......Page 196
  7.4.2 Scaling the Data......Page 198
  7.4.3 DMD in a Weighted Inner Product......Page 199
  7.4.4 The Algorithm......Page 200
  7.4.5 Compressed DMD......Page 201
 7.5 DDMD Based on Rank Revealing QR Factorization......Page 203
  7.5.1 Low-Rank Approximation Using QR Factorization......Page 204
  7.5.2 Implementation Details......Page 206
 7.6 Further Developments and Concluding Remarks......Page 207
 References......Page 210
Part II Methodologies......Page 214
 8.1 Introduction......Page 215
 8.2 Control Problem Formulation......Page 220
  8.3.1 Non-affine Control Systems......Page 222
  8.3.2 Affine Control Systems......Page 225
  8.4.1 Koopman Operator......Page 226
  8.4.2 Perron–Frobenius Operator......Page 230
  8.4.3 Numerical Example......Page 232
  8.5.1 Stability Analysis......Page 233
  8.5.2 Observability and Controllability......Page 236
  8.5.3 Observer Synthesis for State Estimation......Page 237
  8.5.4 Control Design......Page 239
  8.5.5 Sensor and Actuator Placement......Page 241
 8.6 Conclusions and Future Directions......Page 242
 References......Page 243
9 Koopman Model Predictive Control  of Nonlinear Dynamical Systems......Page 253
 9.1 Problem Description......Page 254
 9.2 Koopman Operator for Controlled Systems......Page 255
  9.2.1 Approximation of the Controlled Koopman Operator......Page 256
  9.2.2 Numerical Algorithm—Finding A, B, C......Page 257
 9.3 Koopman MPC......Page 258
  9.3.1 Eliminating Dependence on the Embedding Dimension......Page 260
  9.3.2 Transforming NMPC to Koopman MPC......Page 261
  9.4.1 Input–Output Dynamical Systems......Page 262
  9.4.2 Disturbance/Noise Propagation......Page 264
  9.5.1 Prediction Comparison......Page 265
  9.5.2 Feedback Control of a Bilinear Motor......Page 266
  9.5.3 Nonlinear PDE Control......Page 270
 9.6 Conclusion and Outlook......Page 271
 References......Page 273
 10.1 Introduction......Page 274
 10.2 Reduced Order Modeling Using the Koopman Operator......Page 276
 10.3 Model Predictive Control......Page 279
  10.4.1 Transformation to Switched Systems......Page 280
  10.4.2 Bilinear Models via Linear Interpolation......Page 285
  10.5.1 Test Cases and Reference Setup......Page 288
  10.5.2 Data Sampling and Basis Selection......Page 292
 10.6 Online Updates Using Sensor Data......Page 294
 References......Page 297
 11.1 Introduction......Page 300
  11.2.2 Koopman Operator......Page 303
  11.2.3 Partial Orders and Monotone Systems......Page 304
 11.3 Problem Motivation and Issues with Naive Formulations......Page 306
  11.4.1 Problem Formulation......Page 309
  11.4.2 Existence of Solutions......Page 310
  11.4.3 Pulse Control Function......Page 312
  11.4.4 Solution to the Problem of Convergence to Isostables......Page 314
  11.5.1 Dealing with Parametric Uncertainty......Page 316
  11.5.2 Generalizing the Input Space......Page 317
  11.5.3 Possible Relaxations of the Monotonicity Assumption......Page 318
  11.6.1 Controlling the Generalized Repressilator......Page 320
  11.6.2 HIV Viral Load Control......Page 325
 11.7 Conclusion......Page 327
 References......Page 328
12 Data-Driven Nonlinear Stabilization Using Koopman Operator......Page 330
 12.1 Introduction......Page 331
  12.2.1 Koopman Operator......Page 333
  12.2.2 Fokker–Planck Equation......Page 334
  12.2.3 Feedback Stabilization and Control Lyapunov Functions......Page 336
  12.3.1 Infinite Dimensional Bilinear Representation......Page 338
  12.3.2 Finite-Dimensional Approximation......Page 340
 12.4 Feedback Controller Design......Page 343
 12.5 Simulation Results......Page 345
 References......Page 350
 13.1 Introduction......Page 352
  13.2.2 Koopman Operator and Lifting Method......Page 354
  13.3.1 Description of the Main Method......Page 355
  13.3.2 Algorithm......Page 356
  13.3.3 Variants and Extensions......Page 357
  13.4.1 Description of the Method......Page 359
 13.5 Convergence Results......Page 361
  13.5.1 Convergence of the Main Method......Page 362
  13.5.2 Convergence of the Dual Method......Page 364
 13.6 Examples......Page 366
  13.6.1 Main Method......Page 367
  13.6.2 Dual Method......Page 369
 13.7 Conclusion......Page 371
 References......Page 373
14 Manifold Learning for Data-Driven Dynamical System Analysis......Page 375
 14.1 Introduction......Page 376
 14.2 Background......Page 377
 14.3 The Considered Class of Dynamical Systems......Page 378
 14.4 Data-Driven System Model Inference......Page 380
  14.4.1 New State Representation......Page 381
  14.4.2 State Dynamics......Page 382
  14.4.3 Lift Function......Page 383
 14.5 Diffusion Filtering......Page 384
  14.5.1 Discrete Setting......Page 385
  14.5.2 Contracting Observer......Page 386
  14.5.3 Diffusion Maps Kalman Filter......Page 388
  14.6.1 Tracking Example......Page 389
  14.6.2 Music Analysis......Page 393
 14.7 Conclusion......Page 396
 References......Page 397
15 Phase-Amplitude Reduction of Limit Cycling Systems......Page 399
 15.1 Introduction......Page 400
  15.2.1 Background......Page 402
  15.2.2 Phase Reduction......Page 404
  15.2.3 Entrainment by a Periodic External Input......Page 407
  15.2.4 Optimal Periodic Input for Stable Entrainment......Page 409
  15.2.5 Networks of Coupled Phase Oscillators......Page 410
  15.3.1 Phase, Amplitudes and the Koopman Operator......Page 413
  15.3.2 Phase-Amplitude Reduction......Page 415
  15.3.3 Properties of the Sensitivity Functions......Page 417
  15.3.4 Numerical Methods for the Sensitivity Functions......Page 419
 15.4 Example......Page 421
  15.4.1 Model......Page 422
  15.4.2 Sensitivity Functions of the Phase and Amplitude......Page 423
  15.4.3 Optimal Control Based on the Phase-Amplitude Reduction......Page 424
 15.5 Concluding Remarks......Page 426
 Appendix: Laplace average......Page 427
 References......Page 429
Part III Applications......Page 434
 16.1 Introduction......Page 435
 16.2 Koopman Operator......Page 436
  16.2.1 EDMD Approximation......Page 437
  16.2.2 Incremental Updates......Page 438
  16.2.3 Formulation for Control......Page 439
  16.2.4 Information Measures......Page 440
  16.2.5 Formulation for Hybrid Systems......Page 441
 16.3 Alternative Methods......Page 442
  16.3.1 Generalized Linear Regression......Page 443
  16.3.2 Kernel Ridge Regression......Page 444
  16.3.3 Gaussian Process Regression......Page 446
  16.3.4 Relationships Between Learning Techniques......Page 448
  16.3.5 Example......Page 451
  16.4.1 Learning Sphero SPRK Dynamics in Sand......Page 452
  16.4.2 Rapid Quadcopter Stabilization with Active Learning......Page 454
  16.4.3 Learning SLIP Hybrid Dynamics......Page 457
  16.4.4 Generating Finite Automata via DSS......Page 459
 16.5 Conclusion......Page 461
 References......Page 462
17 Application of Koopman-Based Control in Ultrahigh-Precision Positioning......Page 465
 17.1 Introduction......Page 466
  17.2.2 The LuGre Friction Model......Page 468
  17.2.3 Hsieh's Pre-sliding Friction Model......Page 469
  17.2.4 The GMS Friction Model......Page 470
 17.3 Used Experimental Setups......Page 472
  17.4.1 Friction Parameters of the Rotation Elements......Page 474
  17.4.2 Friction Parameters of the Linear Elements......Page 476
  17.5.1 Proportional-Integral-Derivative (PID) Control......Page 479
  17.5.2 PID Controller with Feedforward Compensation......Page 480
  17.5.3 Model Reference Adaptive Control (MRAC)......Page 481
  17.5.4 Self-tuning Regulator (STR)......Page 483
  17.5.5 Koopman-Based Control......Page 485
 17.6 Conclusions and Outlook......Page 488
 17.7 Notations......Page 490
 References......Page 491
 18.1 Introduction......Page 494
 18.2 Mathematical Formulation......Page 497
 18.3 Modeling of Heat Transfer Dynamics Based on Koopman Mode Decomposition......Page 499
  18.3.1 The Koopman Operator......Page 500
  18.3.2 Koopman Mode Decomposition and Associated Linear Wavenumber Vector......Page 501
  18.4.1 Target Building and Measurement Data......Page 504
  18.4.2 Koopman Mode Decomposition of Measurement Data......Page 508
  18.4.3 Numerical Implementation......Page 511
  18.4.4 Estimation Result......Page 514
 18.5 Possible Applications to Control......Page 515
 18.6 Conclusion......Page 516
 References......Page 517
 19.1 Introduction......Page 520
  19.2.1 Dynamical System and Koopman Operator......Page 522
  19.2.2 Extended Dynamic Mode Decomposition......Page 523
  19.2.3 Delay Embedding with Connections to the Koopman Operator......Page 524
  19.3.1 Power Grid Model......Page 525
  19.3.2 Mathematical Model......Page 526
  19.4.1 Setting......Page 527
  19.4.2 Results......Page 529
 19.5 Concluding Remarks......Page 532
 References......Page 534
20 Koopman Performance Analysis  of Nonlinear Consensus Networks......Page 536
 20.1 Introduction......Page 537
 20.2 Preliminaries......Page 538
 20.3 Koopman Mode Decomposition of System Flows......Page 539
 20.4 Performance of Nonlinear Consensus Networks......Page 542
  20.4.1 Analytic Examples......Page 547
  20.5.1 Smolyak-Collocation Method......Page 553
  20.5.2 Sparse Approximation to Eigenfunctions......Page 556
  20.5.3 Sparse Approximation to Koopman Mode Decomposition......Page 558
  20.5.4 Numerical Examples......Page 560
  20.5.5 Comparison to Extended Dynamic Mode Decomposition......Page 561
 References......Page 562
Index......Page 565