Robust Adaptive Control

Contents......Page 1
Preface......Page 9
List of Acronyms......Page 13
1.2.6 Design of On-Line Parameter Estimators......Page 15
1.2 Adaptive Control......Page 19
1.3 A Brief History......Page 37
2.1 Introduction......Page 40
2.2.1 General Description......Page 41
2.2.2 Canonical State-Space Forms......Page 43
2.3.1 Transfer Functions......Page 48
2.3.2 Coprime Polynomials......Page 53
2.4 Plant Parametric Models......Page 61
2.4.1 Linear Parametric Models......Page 63
2.4.2 Bilinear Parametric Models......Page 72
2.5 Problems......Page 75
3.1 Introduction......Page 80
3.2.1 Norms and Lp Spaces......Page 81
3.2.2 Properties of Functions......Page 86
3.2.3 Positive Definite Matrices......Page 92
3.3.1 Lp Stability......Page 93
3.3.2 The L2± Norm and I/O Stability......Page 99
3.3.3 Small Gain Theorem......Page 110
3.3.4 Bellman-Gronwall Lemma......Page 115
3.4.1 Defnition of Stability......Page 119
3.4.2 Lyapunov's Direct Method......Page 122
3.4.3 Lyapunov-Like Functions......Page 131
3.4.4 Lyapunov's Indirect Method......Page 133
3.4.5 Stability of Linear Systems......Page 134
3.5.1 Positive Real and Strictly Positive Real Transfer Func-
tions......Page 140
3.5.2 PR and SPR Transfer Function Matrices......Page 146
3.6.1 A General LTI Feedback System......Page 148
3.6.2 Internal Stability......Page 149

3.6.3 Sensitivity and Complementary Sensitivity Functions......Page 150
3.6.4 Internal Model Principle......Page 151
3.7 Problems......Page 153
4.1 Introduction......Page 158
4.2.1 Scalar Example: One Unknown Parameter......Page 160
4.2.2 First-Order Example: Two Unknowns......Page 165
4.2.3 Vector Case......Page 170
4.2.4 Remarks......Page 175
4.3.1 Scalar Example......Page 176
4.3.2 First-Order Example......Page 179
4.3.3 General Plant......Page 183
4.3.4 SPR-Lyapunov Design Approach......Page 185
4.3.5 Gradient Method......Page 194
4.3.6 Least-Squares......Page 206
4.3.7 Effect of Initial Conditions......Page 214
4.4.1 Gradient Algorithms with Projection......Page 217
4.4.2 Least-Squares with Projection......Page 220
4.5.1 Known Sign of R¤......Page 222
4.5.2 Sign of ½¤ and Lower Bound ½0 Are Known......Page 226
4.5.3 Unknown Sign of ½¤......Page 229
4.6 Hybrid Adaptive Laws......Page 231
4.8.1 Useful Lemmas......Page 234
4.8.2 Proof of Corollary 4.3.1......Page 249
4.8.3 Proof of Theorem 4.3.2 (iii)......Page 250
4.8.4 Proof of Theorem 4.3.3 (iv)......Page 253
4.8.5 Proof of Theorem 4.3.4 (iv)......Page 254
4.8.6 Proof of Corollary 4.3.2......Page 255
4.8.7 Proof of Theorem 4.5.1(iii)......Page 256
4.8.8 Proof of Theorem 4.6.1 (iii)......Page 257
4.9 Problems......Page 259
5.1 Introduction......Page 264
5.2 Parameter Identifiers......Page 265
5.2.1 Suffciently Rich Signals......Page 266
5.2.2 Parameter Identifiers with Full-State Measurements......Page 272
5.2.3 Parameter Identifiers with Partial-State Measurements......Page 274
5.3.1 The Luenberger Observer......Page 281
5.3.2 The Adaptive Luenberger Observer......Page 283
5.3.3 Hybrid Adaptive Luenberger Observer......Page 290
5.4 Adaptive Observer with Auxiliary Input......Page 293
5.5.1 Adaptive Observer Based on Realization 1......Page 301
5.5.2 Adaptive Observer Based on Realization 2......Page 306
5.6.1 Useful Lemmas......Page 311
5.6.2 Proof of Theorem 5.2.1......Page 315
5.6.3 Proof of Theorem 5.2.2......Page 316
5.6.4 Proof of Theorem 5.2.3......Page 320
5.6.5 Proof of Theorem 5.2.5......Page 323
5.7 Problems......Page 324
6.1 Introduction......Page 327
6.2.1 Scalar Example: Adaptive Regulation......Page 329
6.2.2 Scalar Example: Adaptive Tracking......Page 334
6.2.3 Vector Case: Full-State Measurement......Page 339
6.2.4 Nonlinear Plant......Page 341
6.3 MRC for SISO Plants......Page 344
6.3.1 Problem Statement......Page 345
6.3.2 MRC Schemes: Known Plant Parameters......Page 347
6.4 Direct MRAC with Unnormalized Adaptive
 Laws......Page 357
6.4.1 Relative Degree n¤ = 1......Page 359
6.4.2 Relative Degree n¤ = 2......Page 369
6.4.3 Relative Degree n¤ = 3......Page 376
6.5.1 Example: Adaptive Regulation......Page 386
6.5.2 Example: Adaptive Tracking......Page 393
6.5.3 MRAC for SISO Plants......Page 397
6.5.4 Effect of Initial Conditions......Page 409
6.6 Indirect MRAC......Page 410
6.6.1 Scalar Example......Page 411
6.6.2 Indirect MRAC with Unnormalized Adaptive Laws......Page 415
6.6.3 Indirect MRAC with Normalized Adaptive Law......Page 421
6.7.1 Assumption P1: Minimum Phase......Page 426
6.7.2 Assumption P2: Upper Bound for the Plant Order......Page 427
6.7.3 Assumption P3: Known Relative Degree n¤......Page 428
6.7.4 Tunability......Page 429
6.8.1 Normalizing Properties of Signal mf......Page 431
6.8.2 Proof of Theorem 6.5.1: Direct MRAC......Page 432
6.8.3 Proof of Theorem 6.6.2: Indirect MRAC......Page 438
7.3 PPC: Known Plant Parameters......Page 443
7.1 Introduction......Page 448
7.2.1 Scalar Example: Adaptive Regulation......Page 450
7.2.2 Modified Indirect Adaptive Regulation 
......Page 454
7.2.3 Scalar Example: Adaptive Tracking......Page 456
7.3.1 Problem Statement......Page 462
7.3.2 Polynomial Approach......Page 463
7.3.3 State-Variable Approach......Page 468
7.3.4 Linear Quadratic Control......Page 473
7.4.1 Parametric Model and Adaptive Laws......Page 480
7.4.2 APPC Scheme: The Polynomial Approach......Page 482
7.4.3 APPC Schemes: State-Variable Approach......Page 492
7.4.4 Adaptive Linear Quadratic Control (ALQC)......Page 500
7.5 Hybrid APPC Schemes......Page 508
7.6 Stabilizability Issues and Modified APPC......Page 512
7.6.1 Loss of Stabilizability: A Simple Example......Page 513
7.6.2 Modified APPC Schemes......Page 516
7.6.3 Switched-Excitation Approach......Page 520
7.7.1 Proof of Theorem 7.4.1......Page 527
7.7.2 Proof of Theorem 7.4.2......Page 533
7.7.3 Proof of Theorem 7.5.1......Page 537
7.8 Problems......Page 541
8.1 Introduction......Page 544
8.2 Plant Uncertainties and Robust Control......Page 545
8.2.1 Unstructured Uncertainties......Page 546
8.2.2 Structured Uncertainties: Singular Perturbations......Page 550
8.2.3 Examples of Uncertainty Representations......Page 553
8.2.4 Robust Control......Page 555
8.3 Instability Phenomena in Adaptive Systems......Page 558
8.3.1 Parameter Drift......Page 559
8.3.2 High-Gain Instability......Page 562
8.3.3 Instability Resulting from Fast Adaptation......Page 563
8.3.4 High-Frequency Instability......Page 565
8.3.5 Effect of Parameter Variations......Page 566
8.4 Modifications for Robustness: Simple Examples......Page 568
8.4.1 Leakage......Page 570
8.4.2 Parameter Projection......Page 579
8.4.3 Dead Zone......Page 580
8.4.4 Dynamic Normalization......Page 585
8.5 Robust Adaptive Laws......Page 589
8.5.1 Parametric Models with Modeling Error......Page 590
8.5.2 SPR-Lyapunov Design Approach with Leakage......Page 596
8.5.3 Gradient Algorithms with Leakage......Page 606
8.5.4 Least-Squares with Leakage......Page 616
8.5.5 Projection......Page 617
8.5.6 Dead Zone......Page 620
8.5.7 Bilinear Parametric Model......Page 627
8.5.8 Hybrid Adaptive Laws......Page 630
8.5.9 Effect of Initial Conditions......Page 637
8.6 Summary of Robust Adaptive Laws......Page 638
8.7 Problems......Page 639
9.1 Introduction......Page 648
9.2 Robust Identifiers and Adaptive Observers......Page 649
9.2.1 Dominantly Rich Signals......Page 653
9.2.2 Robust Parameter Identifiers......Page 657
9.2.3 Robust Adaptive Observers......Page 662
9.3 Robust MRAC......Page 664
9.3.1 MRC: Known Plant Parameters......Page 665
9.3.2 Direct MRAC with Unnormalized Adaptive Laws......Page 670
9.3.3 Direct MRAC with Normalized Adaptive Laws......Page 680
9.3.4 Robust Indirect MRAC......Page 701
9.4 Performance Improvement of MRAC......Page 707
9.4.1 Modified MRAC with Unnormalized Adaptive Laws......Page 711
9.4.2 Modified MRAC with Normalized Adaptive Laws......Page 717
9.5 Robust APPC Schemes......Page 723
9.5.1 PPC: Known Parameters......Page 724
9.5.2 Robust Adaptive Laws for APPC Schemes......Page 727
9.5.3 Robust APPC: Polynomial Approach......Page 729
9.5.4 Robust APPC: State Feedback Law......Page 736
9.5.5 Robust LQ Adaptive Control......Page 744
9.6 Adaptive Control of LTV Plants......Page 746
9.7 Adaptive Control for Multivariable Plants......Page 748
9.7.1 Decentralized Adaptive Control......Page 749
9.7.2 The Command Generator Tracker Approach......Page 750
9.7.3 Multivariable MRAC......Page 753
9.8.1 Properties of Fictitious Normalizing Signal......Page 758
9.8.2 Proof of Theorem 9.3.2......Page 762
9.9.1 Proof of Theorem 9.5.2......Page 773
9.9.2 Proof of Theorem 9.5.3......Page 777
9.10 Problems......Page 782
A Swapping Lemmas
......Page 788
B.1 Notation and Mathematical Background......Page 797
B.2 The Method of Steepest Descent (Gradient Method)......Page 799
B.3 Newton's Method......Page 800
B.4 Gradient Projection Method......Page 802
B.5 Example......Page 806
Bibliography......Page 809
Index......Page 832