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robust adaptive control.pdf

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    Contents
    Preface
    List of Acronyms
    1 Introduction
    1.1Control System Design Steps
    1.2 Adaptive Control
    1.2.1 Robust Control
    1.2.2 Gain Scheduling
    1.2.3 Direct and Indirect Adaptive Control
    1.2.4 Model Reference Adaptive Control
    1.2.5 Adaptive Pole Placement Control
    1.2.6 Design of On-Line Parameter Estimators
    1.3 A Brief History
    2 Models for Dynamic Systems
    2.1 Introduction
    2.2 State-Space Models
    2.2.1 General Description
    2.2.2 Canonical State-Space Forms
    2.3 Input/OutputModels
    2.3.1 Transfer Functions
    2.3.2 Coprime Polynomials
    2.4 Plant Parametric Models
    2.4.1 Linear Parametric Models
    2.4.2 Bilinear Parametric Models
    2.5 Problems
    3 Stability
    3.1 Introduction
    3.2 Preliminaries
    3.2.1 Norms and Lp Spaces
    3.2.2 Properties of Functions
    3.2.3 Positive Definite Matrices
    Input/Output Stability
    3.3.1 Lp Stability
    3.3.2 The L2± Norm and I/O Stability
    3.3.3 Small Gain Theorem
    3.3.4 Bellman-Gronwall Lemma
    3.4 Lyapunov Stability
    3.4.1 Defnition of Stability
    3.4.2 Lyapunov's Direct Method
    3.4.3 Lyapunov-Like Functions
    3.4.4 Lyapunov's Indirect Method
    3.4.5 Stability of Linear Systems
    3.5 Positive Real Functions and Stability
    3.5.1 Positive Real and Strictly Positive Real Transfer Func- tions
    3.5.2 PR and SPR Transfer Function Matrices
    3.6 Stability of LTI Feedback Systems
    3.6.1 A General LTI Feedback System
    3.6.2 Internal Stability
    3.6.3 Sensitivity and Complementary Sensitivity Functions
    3.6.4 Internal Model Principle
    3.7 Problems
    4 On-Line Parameter Estimation
    4.1 Introduction
    4.2 Simple Examples
    4.2.1 Scalar Example: One Unknown Parameter
    4.2.2 First-Order Example: Two Unknowns
    4.2.3 Vector Case
    4.2.4 Remarks
    4.3 Adaptive Laws with Normalization
    4.3.1 Scalar Example
    4.3.2 First-Order Example
    4.3.3 General Plant
    4.3.4 SPR-Lyapunov Design Approach
    4.3.5 Gradient Method
    4.3.6 Least-Squares
    4.3.7 Effect of Initial Conditions
    4.4 Adaptive Laws with Projection
    4.4.1 Gradient Algorithms with Projection
    4.4.2 Least-Squares with Projection
    4.5 Bilinear Parametric Model
    4.5.1 Known Sign of R¤
    4.5.2 Sign of ½¤ and Lower Bound ½0 Are Known
    4.5.3 Unknown Sign of ½¤
    4.6 Hybrid Adaptive Laws
    4.7 Summary of Adaptive Laws
    4.8 Parameter Convergence Proofs
    4.8.1 Useful Lemmas
    4.8.2 Proof of Corollary 4.3.1
    4.8.3 Proof of Theorem 4.3.2 (iii)
    4.8.4 Proof of Theorem 4.3.3 (iv)
    4.8.5 Proof of Theorem 4.3.4 (iv)
    4.8.6 Proof of Corollary 4.3.2
    4.8.7 Proof of Theorem 4.5.1(iii)
    4.8.8 Proof of Theorem 4.6.1 (iii)
    4.9 Problems
    5 Parameter Identifiers and Adaptive Observers
    5.1 Introduction
    5.2 Parameter Identifiers
    5.2.1 Suffciently Rich Signals
    5.2.2 Parameter Identifiers with Full-State Measurements
    5.2.3 Parameter Identifiers with Partial-State Measurements
    5.3 Adaptive Observers
    5.3.1 The Luenberger Observer
    5.3.2 The Adaptive Luenberger Observer
    5.3.3 Hybrid Adaptive Luenberger Observer
    5.4 Adaptive Observer with Auxiliary Input
    5.5 Adaptive Observers for Nonminimal Plant Models
    5.5.1 Adaptive Observer Based on Realization 1
    5.5.2 Adaptive Observer Based on Realization 2
    5.6 Parameter Convergence Proofs
    5.6.1 Useful Lemmas
    5.6.2 Proof of Theorem 5.2.1
    5.6.3 Proof of Theorem 5.2.2
    5.6.4 Proof of Theorem 5.2.3
    5.6.5 Proof of Theorem 5.2.5
    5.7 Problems
    6 Model Reference Adaptive Control
    6.1 Introduction
    6.2 Simple Direct MRAC Schemes
    6.2.1 Scalar Example: Adaptive Regulation
    6.2.2 Scalar Example: Adaptive Tracking
    6.2.3 Vector Case: Full-State Measurement
    6.2.4 Nonlinear Plant
    6.3 MRC for SISO Plants
    6.3.1 Problem Statement
    6.3.2 MRC Schemes: Known Plant Parameters
    6.4 Direct MRAC with Unnormalized Adaptive Laws
    6.4.1 Relative Degree n¤ = 1
    6.4.2 Relative Degree n¤ = 2
    6.4.3 Relative Degree n¤ = 3
    6.5 Direct MRAC with Normalized Adaptive Laws
    6.5.1 Example: Adaptive Regulation
    6.5.2 Example: Adaptive Tracking
    6.5.3 MRAC for SISO Plants
    6.5.4 Effect of Initial Conditions
    6.6 Indirect MRAC
    6.6.1 Scalar Example
    6.6.2 Indirect MRAC with Unnormalized Adaptive Laws
    6.6.3 Indirect MRAC with Normalized Adaptive Law
    6.7 Relaxation of Assumptions in MRAC
    6.7.1 Assumption P1: Minimum Phase
    6.7.2 Assumption P2: Upper Bound for the Plant Order
    6.7.3 Assumption P3: Known Relative Degree n¤
    6.7.4 Tunability
    6.8 Stability Proofs of MRAC Schemes
    6.8.1 Normalizing Properties of Signal mf
    6.8.2 Proof of Theorem 6.5.1: Direct MRAC
    6.8.3 Proof of Theorem 6.6.2: Indirect MRAC
    6.9 Problems
    7 Adaptive Pole Placement Control
    7.1 Introduction
    7.2 Simple APPC Schemes
    7.2.1 Scalar Example: Adaptive Regulation
    7.2.2 Modified Indirect Adaptive Regulation
    7.2.3 Scalar Example: Adaptive Tracking
    7.3 PPC: Known Plant Parameters
    7.3.1 Problem Statement
    7.3.2 Polynomial Approach
    7.3.3 State-Variable Approach
    7.3.4 Linear Quadratic Control
    7.4 Indirect APPC Schemes
    7.4.1 Parametric Model and Adaptive Laws
    7.4.2 APPC Scheme: The Polynomial Approach
    7.4.3 APPC Schemes: State-Variable Approach
    7.4.4 Adaptive Linear Quadratic Control (ALQC)
    7.5 Hybrid APPC Schemes
    7.6 Stabilizability Issues and Modified APPC
    7.6.1 Loss of Stabilizability: A Simple Example
    7.6.2 Modified APPC Schemes
    7.6.3 Switched-Excitation Approach
    7.7 Stability Proofs
    7.7.1 Proof of Theorem 7.4.1
    7.7.2 Proof of Theorem 7.4.2
    7.7.3 Proof of Theorem 7.5.1
    7.8 Problems
    8 Robust Adaptive Laws
    8.1 Introduction
    8.2 Plant Uncertainties and Robust Control
    8.2.1 Unstructured Uncertainties
    8.2.2 Structured Uncertainties: Singular Perturbations
    8.2.3 Examples of Uncertainty Representations
    8.2.4 Robust Control
    8.3 Instability Phenomena in Adaptive Systems
    8.3.1 Parameter Drift
    8.3.2 High-Gain Instability
    8.3.3 Instability Resulting from Fast Adaptation
    8.3.4 High-Frequency Instability
    8.3.5 Effect of Parameter Variations
    8.4 Modifications for Robustness: Simple Examples
    8.4.1 Leakage
    8.4.2 Parameter Projection
    8.4.3 Dead Zone
    8.4.4 Dynamic Normalization
    8.5 Robust Adaptive Laws
    8.5.1 Parametric Models with Modeling Error
    8.5.2 SPR-Lyapunov Design Approach with Leakage
    8.5.3 Gradient Algorithms with Leakage
    8.5.4 Least-Squares with Leakage
    8.5.5 Projection
    8.5.6 Dead Zone
    8.5.7 Bilinear Parametric Model
    8.5.8 Hybrid Adaptive Laws
    8.5.9 Effect of Initial Conditions
    8.6 Summary of Robust Adaptive Laws
    8.7 Problems
    9 Robust Adaptive Control Schemes
    9.1 Introduction
    9.2 Robust Identifiers and Adaptive Observers
    9.2.1 Dominantly Rich Signals
    9.2.2 Robust Parameter Identifiers
    9.2.3 Robust Adaptive Observers
    9.3 Robust MRAC
    9.3.1 MRC: Known Plant Parameters
    9.3.2 Direct MRAC with Unnormalized Adaptive Laws
    9.3.3 Direct MRAC with Normalized Adaptive Laws
    9.3.4 Robust Indirect MRAC
    9.4 Performance Improvement of MRAC
    9.4.1 Modified MRAC with Unnormalized Adaptive Laws
    9.4.2 Modified MRAC with Normalized Adaptive Laws
    9.5 Robust APPC Schemes
    9.5.1 PPC: Known Parameters
    9.5.2 Robust Adaptive Laws for APPC Schemes
    9.5.3 Robust APPC: Polynomial Approach
    9.5.4 Robust APPC: State Feedback Law
    9.5.5 Robust LQ Adaptive Control
    9.6 Adaptive Control of LTV Plants
    9.7 Adaptive Control for Multivariable Plants
    9.7.1 Decentralized Adaptive Control
    9.7.2 The Command Generator Tracker Approach
    9.7.3 Multivariable MRAC
    9.8 Stability Proofs of Robust MRAC Schemes
    9.8.1 Properties of Fictitious Normalizing Signal
    9.8.2 Proof of Theorem 9.3.2
    9.9 Stability Proofs of Robust APPC Schemes
    9.9.1 Proof of Theorem 9.5.2
    9.9.2 Proof of Theorem 9.5.3
    9.10 Problems
    Appendix
    A Swapping Lemmas
    B Optimization Techniques
    B.1 Notation and Mathematical Background
    B.2 The Method of Steepest Descent (Gradient Method)
    B.3 Newton's Method
    B.4 Gradient Projection Method
    B.5 Example
    Bibliography
    Index
    Robust Adptive Control Robust Adptive Control
    Contents Preface List of Acronyms 1 Introduction . . . . . . . . . . . . . . . . . . 1.1 Control System Design Steps 1.2 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Robust Control . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Direct and Indirect Adaptive Control 1.2.4 Model Reference Adaptive Control . . . . . . . . . . . 1.2.5 Adaptive Pole Placement Control . . . . . . . . . . . . 1.2.6 Design of On-Line Parameter Estimators . . . . . . . 1.3 A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Models for Dynamic Systems 2.1 2.2 State-Space Models Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 General Description . . . . . . . . . . . . . . . . . . . 2.2.2 Canonical State-Space Forms . . . . . . . . . . . . . . Input/Output Models . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Transfer Functions . . . . . . . . . . . . . . . . . . . . 2.3.2 Coprime Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Linear Parametric Models . . . . . . . . . . . . . . . . 2.4.2 Bilinear Parametric Models . . . . . . . . . . . . . . . 2.3 2.4 Plant Parametric Models xiii xvii 1 1 5 6 7 8 12 14 16 23 26 26 27 27 29 34 34 39 47 49 58 v
    vi CONTENTS 2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 Stability 3.3 3.1 3.2 Preliminaries 66 66 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Norms and Lp Spaces 67 . . . . . . . . . . . . . . . . . . 72 3.2.2 Properties of Functions . . . . . . . . . . . . . . . . . 78 3.2.3 Positive Definite Matrices . . . . . . . . . . . . . . . . 79 Input/Output Stability . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Lp Stability . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.2 The L2δ Norm and I/O Stability . . . . . . . . . . . . 85 96 Small Gain Theorem . . . . . . . . . . . . . . . . . . . 3.3.3 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 . . . . . . . . . . . . . . . 120 3.5 Positive Real Functions and Stability . . . . . . . . . . . . . . 126 Stability of Linear Systems 3.5.1 Positive Real and Strictly Positive Real Transfer Func- 3.6 Stability of LTI Feedback Systems 3.5.2 PR and SPR Transfer Function Matrices tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 . . . . . . . 132 . . . . . . . . . . . . . . . 134 3.6.1 A General LTI Feedback System . . . . . . . . . . . . 134 3.6.2 Internal Stability . . . . . . . . . . . . . . . . . . . . . 135 Sensitivity and Complementary Sensitivity Functions . 136 3.6.3 Internal Model Principle . . . . . . . . . . . . . . . . . 137 3.6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.7 Problems 4 On-Line Parameter Estimation 144 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.2 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2.1 . . . . . . 146 4.2.2 First-Order Example: Two Unknowns . . . . . . . . . 151 4.2.3 Vector Case . . . . . . . . . . . . . . . . . . . . . . . . 156 Scalar Example: One Unknown Parameter
    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 Sign of ρ∗ and Lower Bound ρ0 Are Known . . . . . . 212 4.5.2 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.9 Problems 5 Parameter Identifiers and Adaptive Observers 250 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.2 Parameter Identifiers . . . . . . . . . . . . . . . . . . . . . . . 251 Sufficiently Rich Signals . . . . . . . . . . . . . . . . . 252 5.2.1 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 . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 5.6 Parameter Convergence Proofs 5.7 Problems 6 Model Reference Adaptive Control 6.6 6.3 MRC for SISO Plants 6.1 6.2 Simple Direct MRAC Schemes 313 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 . . . . . . . . . . . . . . . . . 315 Scalar Example: Adaptive Regulation . . . . . . . . . 315 6.2.1 Scalar Example: Adaptive Tracking . . . . . . . . . . 320 6.2.2 6.2.3 Vector Case: Full-State Measurement . . . . . . . . . 325 6.2.4 Nonlinear Plant . . . . . . . . . . . . . . . . . . . . . . 328 . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . 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 Indirect MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . 397 6.6.1 Scalar Example . . . . . . . . . . . . . . . . . . . . . . 398 6.5 Direct MRAC with Normalized Adaptive Laws
    CONTENTS ix 6.6.2 6.6.3 Indirect MRAC with Unnormalized Adaptive Laws . . 402 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 6.9 Problems 7 Adaptive Pole Placement Control 7.4 435 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 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 Switched-Excitation Approach . . . . . . . . . . . . . 507 7.6.3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 7.8 Problems 8 Robust Adaptive Laws 8.3 8.1 8.2 Plant Uncertainties and Robust Control 531 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 . . . . . . . . . . . . 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 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 . . . . . . . . . . . . . . . 624 8.5.9 Effect of Initial Conditions . . . . . . . . . . . . . . 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 8.6 Summary of Robust Adaptive Laws 8.7 Problems
    CONTENTS xi 9 Robust Adaptive Control Schemes 9.5 Robust APPC 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 . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . 731 9.5.5 Robust LQ Adaptive Control 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 . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 Swapping Lemmas . . . . . . . . . . . . . . . . . . . . . . . . 775 Optimization Techniques . . . . . . . . . . . . . . . . . . . . . 784 Notation and Mathematical Background . . . . . . . . 784 B.1 B.2 The Method of Steepest Descent (Gradient Method) . 786 9.8 Stability Proofs of Robust MRAC Schemes 9.10 Problems A B
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