Contents
Preface
xiii
List of Acronyms
xvii
1 Introduction
1
1.1 Control System Design Steps . . . . . . . . . . . . . . . . . .
1
1.2 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1
Robust Control . . . . . . . . . . . . . . . . . . . . . .
6
1.2.2
Gain Scheduling . . . . . . . . . . . . . . . . . . . . .
7
1.2.3
Direct and Indirect Adaptive Control . . . . . . . . .
8
1.2.4
Model Reference Adaptive Control . . . . . . . . . . .
12
1.2.5
Adaptive Pole Placement Control . . . . . . . . . . . .
14
1.2.6
Design of On-Line Parameter Estimators . . . . . . .
16
1.3 A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2 Models for Dynamic Systems
26
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.2 State-Space Models
. . . . . . . . . . . . . . . . . . . . . . .
27
2.2.1
General Description . . . . . . . . . . . . . . . . . . .
27
2.2.2
Canonical State-Space Forms . . . . . . . . . . . . . .
29
2.3 Input/Output Models . . . . . . . . . . . . . . . . . . . . . .
34
2.3.1
Transfer Functions . . . . . . . . . . . . . . . . . . . .
34
2.3.2
Coprime Polynomials . . . . . . . . . . . . . . . . . .
39
2.4 Plant Parametric Models . . . . . . . . . . . . . . . . . . . .
47
2.4.1
Linear Parametric Models . . . . . . . . . . . . . . . .
49
2.4.2
Bilinear Parametric Models . . . . . . . . . . . . . . .
58
v
vi
CONTENTS
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3 Stability
66
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.2.1
Norms and Lp Spaces . . . . . . . . . . . . . . . . . . 67
3.2.2
Properties of Functions . . . . . . . . . . . . . . . . .
72
3.2.3
Positive Definite Matrices . . . . . . . . . . . . . . . .
78
3.3 Input/Output Stability . . . . . . . . . . . . . . . . . . . . . .
79
3.3.1
Lp Stability . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.2
The L 2 δ Norm and I/O Stability . . . . . . . . . . . . 85
3.3.3
Small Gain Theorem . . . . . . . . . . . . . . . . . . .
96
3.3.4
Bellman-Gronwall Lemma . . . . . . . . . . . . . . . . 101
3.4 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4.1
Definition of Stability . . . . . . . . . . . . . . . . . . 105
3.4.2
Lyapunov’s Direct Method . . . . . . . . . . . . . . . 108
3.4.3
Lyapunov-Like Functions . . . . . . . . . . . . . . . . 117
3.4.4
Lyapunov’s Indirect Method . . . . . . . . . . . . . . . 119
3.4.5
Stability of Linear Systems . . . . . . . . . . . . . . . 120
3.5 Positive Real Functions and Stability . . . . . . . . . . . . . . 126
3.5.1
Positive Real and Strictly Positive Real Transfer Func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.5.2
PR and SPR Transfer Function Matrices . . . . . . . 132
3.6 Stability of LTI Feedback Systems . . . . . . . . . . . . . . . 134
3.6.1
A General LTI Feedback System . . . . . . . . . . . . 134
3.6.2
Internal Stability . . . . . . . . . . . . . . . . . . . . . 135
3.6.3
Sensitivity and Complementary Sensitivity Functions . 136
3.6.4
Internal Model Principle . . . . . . . . . . . . . . . . . 137
3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4 On-Line Parameter Estimation
144
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.2 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.2.1
Scalar Example: One Unknown Parameter . . . . . . 146
4.2.2
First-Order Example: Two Unknowns . . . . . . . . . 151
4.2.3
Vector Case . . . . . . . . . . . . . . . . . . . . . . . . 156
CONTENTS
vii
4.2.4
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.3 Adaptive Laws with Normalization . . . . . . . . . . . . . . . 162
4.3.1
Scalar Example . . . . . . . . . . . . . . . . . . . . . . 162
4.3.2
First-Order Example . . . . . . . . . . . . . . . . . . . 165
4.3.3
General Plant . . . . . . . . . . . . . . . . . . . . . . . 169
4.3.4
SPR-Lyapunov Design Approach . . . . . . . . . . . . 171
4.3.5
Gradient Method . . . . . . . . . . . . . . . . . . . . . 180
4.3.6
Least-Squares . . . . . . . . . . . . . . . . . . . . . . . 192
4.3.7
Effect of Initial Conditions . . . . . . . . . . . . . . . 200
4.4 Adaptive Laws with Projection . . . . . . . . . . . . . . . . . 203
4.4.1
Gradient Algorithms with Projection . . . . . . . . . . 203
4.4.2
Least-Squares with Projection . . . . . . . . . . . . . . 206
4.5 Bilinear Parametric Model . . . . . . . . . . . . . . . . . . . . 208
4.5.1
Known Sign of ρ∗ . . . . . . . . . . . . . . . . . . . . . 208
4.5.2
Sign of ρ∗ and Lower Bound ρ 0 Are Known . . . . . . 212
4.5.3
Unknown Sign of ρ∗ . . . . . . . . . . . . . . . . . . . 215
4.6 Hybrid Adaptive Laws . . . . . . . . . . . . . . . . . . . . . . 217
4.7 Summary of Adaptive Laws . . . . . . . . . . . . . . . . . . . 220
4.8 Parameter Convergence Proofs . . . . . . . . . . . . . . . . . 220
4.8.1
Useful Lemmas . . . . . . . . . . . . . . . . . . . . . . 220
4.8.2
Proof of Corollary 4.3.1 . . . . . . . . . . . . . . . . . 235
4.8.3
Proof of Theorem 4.3.2 (iii) . . . . . . . . . . . . . . . 236
4.8.4
Proof of Theorem 4.3.3 (iv) . . . . . . . . . . . . . . . 239
4.8.5
Proof of Theorem 4.3.4 (iv) . . . . . . . . . . . . . . . 240
4.8.6
Proof of Corollary 4.3.2 . . . . . . . . . . . . . . . . . 241
4.8.7
Proof of Theorem 4.5.1(iii) . . . . . . . . . . . . . . . 242
4.8.8
Proof of Theorem 4.6.1 (iii) . . . . . . . . . . . . . . . 243
4.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
5 Parameter Identifiers and Adaptive Observers
250
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.2 Parameter Identifiers . . . . . . . . . . . . . . . . . . . . . . . 251
5.2.1
Sufficiently Rich Signals . . . . . . . . . . . . . . . . . 252
5.2.2
Parameter Identifiers with Full-State Measurements . 258
5.2.3
Parameter Identifiers with Partial-State Measurements 260
5.3 Adaptive Observers . . . . . . . . . . . . . . . . . . . . . . . . 267
viii
CONTENTS
5.3.1
The Luenberger Observer . . . . . . . . . . . . . . . . 267
5.3.2
The Adaptive Luenberger Observer . . . . . . . . . . . 269
5.3.3
Hybrid Adaptive Luenberger Observer . . . . . . . . . 276
5.4
Adaptive Observer with Auxiliary Input . . . . . . . . . . . 279
5.5 Adaptive Observers for Nonminimal Plant Models . . . . . 287
5.5.1
Adaptive Observer Based on Realization 1 . . . . . . . 287
5.5.2
Adaptive Observer Based on Realization 2 . . . . . . . 292
5.6 Parameter Convergence Proofs . . . . . . . . . . . . . . . . . 297
5.6.1
Useful Lemmas . . . . . . . . . . . . . . . . . . . . . . 297
5.6.2
Proof of Theorem 5.2.1 . . . . . . . . . . . . . . . . . 301
5.6.3
Proof of Theorem 5.2.2 . . . . . . . . . . . . . . . . . 302
5.6.4
Proof of Theorem 5.2.3 . . . . . . . . . . . . . . . . . 306
5.6.5
Proof of Theorem 5.2.5 . . . . . . . . . . . . . . . . . 309
5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
6 Model Reference Adaptive Control
313
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
6.2 Simple Direct MRAC Schemes . . . . . . . . . . . . . . . . . 315
6.2.1
Scalar Example: Adaptive Regulation . . . . . . . . . 315
6.2.2
Scalar Example: Adaptive Tracking . . . . . . . . . . 320
6.2.3
Vector Case: Full-State Measurement . . . . . . . . . 325
6.2.4
Nonlinear Plant . . . . . . . . . . . . . . . . . . . . . . 328
6.3 MRC for SISO Plants . . . . . . . . . . . . . . . . . . . . . . 330
6.3.1
Problem Statement . . . . . . . . . . . . . . . . . . . . 331
6.3.2
MRC Schemes: Known Plant Parameters . . . . . . . 333
6.4 Direct MRAC with Unnormalized Adaptive Laws . . . . . . . 344
6.4.1
Relative Degree n∗ = 1 . . . . . . . . . . . . . . . . . 345
6.4.2
Relative Degree n∗ = 2 . . . . . . . . . . . . . . . . . 356
6.4.3
Relative Degree n∗ = 3 . . . . . . . . . . . . . . . . . . 363
6.5 Direct MRAC with Normalized Adaptive Laws . . . . . . . 373
6.5.1
Example: Adaptive Regulation . . . . . . . . . . . . . 373
6.5.2
Example: Adaptive Tracking . . . . . . . . . . . . . . 380
6.5.3
MRAC for SISO Plants . . . . . . . . . . . . . . . . . 384
6.5.4
Effect of Initial Conditions . . . . . . . . . . . . . . . 396
6.6 Indirect MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . 397
6.6.1
Scalar Example . . . . . . . . . . . . . . . . . . . . . . 398
CONTENTS
ix
6.6.2
Indirect MRAC with Unnormalized Adaptive Laws . . 402
6.6.3
Indirect MRAC with Normalized Adaptive Law . . . . 408
6.7 Relaxation of Assumptions in MRAC . . . . . . . . . . . . . . 413
6.7.1
Assumption P1: Minimum Phase . . . . . . . . . . . . 413
6.7.2
Assumption P2: Upper Bound for the Plant Order . . 414
6.7.3
Assumption P3: Known Relative Degree n∗ . . . . . . 415
6.7.4
Tunability . . . . . . . . . . . . . . . . . . . . . . . . . 416
6.8 Stability Proofs of MRAC Schemes . . . . . . . . . . . . . . . 418
6.8.1
Normalizing Properties of Signal mf . . . . . . . . . . 418
6.8.2
Proof of Theorem 6.5.1: Direct MRAC . . . . . . . . . 419
6.8.3
Proof of Theorem 6.6.2: Indirect MRAC . . . . . . . . 425
6.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
7 Adaptive Pole Placement Control
435
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
7.2 Simple APPC Schemes . . . . . . . . . . . . . . . . . . . . . . 437
7.2.1
Scalar Example: Adaptive Regulation . . . . . . . . . 437
7.2.2
Modified Indirect Adaptive Regulation . . . . . . . . . 441
7.2.3
Scalar Example: Adaptive Tracking . . . . . . . . . . 443
7.3 PPC: Known Plant Parameters . . . . . . . . . . . . . . . . . 448
7.3.1
Problem Statement . . . . . . . . . . . . . . . . . . . . 449
7.3.2
Polynomial Approach . . . . . . . . . . . . . . . . . . 450
7.3.3
State-Variable Approach . . . . . . . . . . . . . . . . . 455
7.3.4
Linear Quadratic Control . . . . . . . . . . . . . . . . 460
7.4 Indirect APPC Schemes . . . . . . . . . . . . . . . . . . . . . 467
7.4.1
Parametric Model and Adaptive Laws . . . . . . . . . 467
7.4.2
APPC Scheme: The Polynomial Approach . . . . . . . 469
7.4.3
APPC Schemes: State-Variable Approach . . . . . . . 479
7.4.4
Adaptive Linear Quadratic Control (ALQC) . . . . . 487
7.5 Hybrid APPC Schemes . . . . . . . . . . . . . . . . . . . . . 495
7.6 Stabilizability Issues and Modified APPC . . . . . . . . . . . 499
7.6.1
Loss of Stabilizability: A Simple Example . . . . . . . 500
7.6.2
Modified APPC Schemes . . . . . . . . . . . . . . . . 503
7.6.3
Switched-Excitation Approach . . . . . . . . . . . . . 507
7.7 Stability Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 514
7.7.1
Proof of Theorem 7.4.1 . . . . . . . . . . . . . . . . . 514
x
CONTENTS
7.7.2
Proof of Theorem 7.4.2 . . . . . . . . . . . . . . . . . 520
7.7.3
Proof of Theorem 7.5.1 . . . . . . . . . . . . . . . . . 524
7.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
8 Robust Adaptive Laws
531
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
8.2 Plant Uncertainties and Robust Control . . . . . . . . . . . . 532
8.2.1
Unstructured Uncertainties . . . . . . . . . . . . . . . 533
8.2.2
Structured Uncertainties: Singular Perturbations . . . 537
8.2.3
Examples of Uncertainty Representations . . . . . . . 540
8.2.4
Robust Control . . . . . . . . . . . . . . . . . . . . . . 542
8.3 Instability Phenomena in Adaptive Systems . . . . . . . . . . 545
8.3.1
Parameter Drift . . . . . . . . . . . . . . . . . . . . . 546
8.3.2
High-Gain Instability . . . . . . . . . . . . . . . . . . 549
8.3.3
Instability Resulting from Fast Adaptation . . . . . . 550
8.3.4
High-Frequency Instability . . . . . . . . . . . . . . . 552
8.3.5
Effect of Parameter Variations . . . . . . . . . . . . . 553
8.4 Modifications for Robustness: Simple Examples . . . . . . . . 555
8.4.1
Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . 557
8.4.2
Parameter Projection . . . . . . . . . . . . . . . . . . 566
8.4.3
Dead Zone . . . . . . . . . . . . . . . . . . . . . . . . 567
8.4.4
Dynamic Normalization . . . . . . . . . . . . . . . . . 572
8.5 Robust Adaptive Laws . . . . . . . . . . . . . . . . . . . . . . 576
8.5.1
Parametric Models with Modeling Error . . . . . . . . 577
8.5.2
SPR-Lyapunov Design Approach with Leakage . . . . 583
8.5.3
Gradient Algorithms with Leakage . . . . . . . . . . . 593
8.5.4
Least-Squares with Leakage . . . . . . . . . . . . . . . 603
8.5.5
Projection . . . . . . . . . . . . . . . . . . . . . . . . . 604
8.5.6
Dead Zone . . . . . . . . . . . . . . . . . . . . . . . . 607
8.5.7
Bilinear Parametric Model . . . . . . . . . . . . . . . . 614
8.5.8
Hybrid Adaptive Laws . . . . . . . . . . . . . . . . . . 617
8.5.9
Effect of Initial Conditions . . . . . . . . . . . . . . . 624
8.6 Summary of Robust Adaptive Laws . . . . . . . . . . . . . . 624
8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
CONTENTS
xi
9 Robust Adaptive Control Schemes
635
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
9.2 Robust Identifiers and Adaptive Observers . . . . . . . . . . . 636
9.2.1
Dominantly Rich Signals . . . . . . . . . . . . . . . . . 639
9.2.2
Robust Parameter Identifiers . . . . . . . . . . . . . . 644
9.2.3
Robust Adaptive Observers . . . . . . . . . . . . . . . 649
9.3 Robust MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . 651
9.3.1
MRC: Known Plant Parameters . . . . . . . . . . . . 652
9.3.2
Direct MRAC with Unnormalized Adaptive Laws . . . 657
9.3.3
Direct MRAC with Normalized Adaptive Laws . . . . 667
9.3.4
Robust Indirect MRAC . . . . . . . . . . . . . . . . . 688
9.4 Performance Improvement of MRAC . . . . . . . . . . . . . . 694
9.4.1
Modified MRAC with Unnormalized Adaptive Laws . 698
9.4.2
Modified MRAC with Normalized Adaptive Laws . . . 704
9.5 Robust APPC Schemes . . . . . . . . . . . . . . . . . . . . . 710
9.5.1
PPC: Known Parameters . . . . . . . . . . . . . . . . 711
9.5.2
Robust Adaptive Laws for APPC Schemes . . . . . . . 714
9.5.3
Robust APPC: Polynomial Approach . . . . . . . . . 716
9.5.4
Robust APPC: State Feedback Law . . . . . . . . . . 723
9.5.5
Robust LQ Adaptive Control . . . . . . . . . . . . . . 731
9.6 Adaptive Control of LTV Plants . . . . . . . . . . . . . . . . 733
9.7 Adaptive Control for Multivariable Plants . . . . . . . . . . . 735
9.7.1
Decentralized Adaptive Control . . . . . . . . . . . . . 736
9.7.2
The Command Generator Tracker Approach . . . . . 737
9.7.3
Multivariable MRAC . . . . . . . . . . . . . . . . . . . 740
9.8 Stability Proofs of Robust MRAC Schemes . . . . . . . . . . 745
9.8.1
Properties of Fictitious Normalizing Signal . . . . . . 745
9.8.2
Proof of Theorem 9.3.2 . . . . . . . . . . . . . . . . . 749
9.9 Stability Proofs of Robust APPC Schemes . . . . . . . . . . . 760
9.9.1
Proof of Theorem 9.5.2 . . . . . . . . . . . . . . . . . 760
9.9.2
Proof of Theorem 9.5.3 . . . . . . . . . . . . . . . . . 764
9.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
A
Swapping Lemmas . . . . . . . . . . . . . . . . . . . . . . . . 775
B
Optimization Techniques . . . . . . . . . . . . . . . . . . . . . 784
B.1
Notation and Mathematical Background . . . . . . . . 784
B.2
The Method of Steepest Descent (Gradient Method) . 786
xii
CONTENTS
B.3
Newton’s Method . . . . . . . . . . . . . . . . . . . . . 787
B.4
Gradient Projection Method . . . . . . . . . . . . . . 789
B.5
Example . . . . . . . . . . . . . . . . . . . . . . . . . . 792
Bibliography
796
Index
819
License Agreement and Limited Warranty
822
Preface
The area of adaptive control has grown to be one of the richest in terms of
algorithms, design techniques, analytical tools, and modifications. Several
books and research monographs already exist on the topics of parameter
estimation and adaptive control.
Despite this rich literature, the field of adaptive control may easily appear
to an outsider as a collection of unrelated tricks and modifications. Students
are often overwhelmed and sometimes confused by the vast number of what
appear to be unrelated designs and analytical methods achieving similar re-
sults. Researchers concentrating on different approaches in adaptive control
often find it difficult to relate their techniques with others without additional
research efforts.
The purpose of this book is to alleviate some of the confusion and diffi-
culty in understanding the design, analysis, and robustness of a wide class
of adaptive control for continuous-time plants. The book is the outcome of
several years of research, whose main purpose was not to generate new re-
sults, but rather unify, simplify, and present in a tutorial manner most of the
existing techniques for designing and analyzing adaptive control systems.
The book is written in a self-contained fashion to be used as a textbook
on adaptive systems at the senior undergraduate, or first and second gradu-
ate level. It is assumed that the reader is familiar with the materials taught
in undergraduate courses on linear systems, differential equations, and auto-
matic control. The book is also useful for an industrial audience where the
interest is to implement adaptive control rather than analyze its stability
properties. Tables with descriptions of adaptive control schemes presented
in the book are meant to serve this audience. The personal computer floppy
disk, included with the book, provides several examples of simple adaptive
xiii
xiv
PREFACE
control systems that will help the reader understand some of the implemen-
tation aspects of adaptive systems.
A significant part of the book, devoted to parameter estimation and
learning in general, provides techniques and algorithms for on-line fitting
of dynamic or static models to data generated by real systems. The tools
for design and analysis presented in the book are very valuable in under-
standing and analyzing similar parameter estimation problems that appear
in neural networks, fuzzy systems, and other universal approximators. The
book will be of great interest to the neural and fuzzy logic audience who
will benefit from the strong similarity that exists between adaptive systems,
whose stability properties are well established, and neural networks, fuzzy
logic systems where stability and convergence issues are yet to be resolved.
The book is organized as follows: Chapter 1 is used to introduce adap-
tive control as a method for controlling plants with parametric uncertainty.
It also provides some background and a brief history of the development
of adaptive control. Chapter 2 presents a review of various plant model
representations that are useful for parameter identification and control. A
considerable number of stability results that are useful in analyzing and un-
derstanding the properties of adaptive and nonlinear systems in general are
presented in Chapter 3. Chapter 4 deals with the design and analysis of on-
line parameter estimators or adaptive laws that form the backbone of every
adaptive control scheme presented in the chapters to follow. The design of
parameter identifiers and adaptive observers for stable plants is presented
in Chapter 5. Chapter 6 is devoted to the design and analysis of a wide
class of model reference adaptive controllers for minimum phase plants. The
design of adaptive control for plants that are not necessarily minimum phase
is presented in Chapter 7. These schemes are based on pole placement con-
trol strategies and are referred to as adaptive pole placement control. While
Chapters 4 through 7 deal with plant models that are free of disturbances,
unmodeled dynamics and noise, Chapters 8 and 9 deal with the robustness
issues in adaptive control when plant model uncertainties, such as bounded
disturbances and unmodeled dynamics, are present.