System Identification with Quantized Observations

by Le Yi Wang, G. George Yin, Ji-Feng Zhang, Yanlong Zhao

Overview

This book concerns the identification of systems in which only quantized output observations are available, due to sensor limitations, signal quan- zation, or coding for communications. Although there are many excellent treaties in system identification and its related subject areas, a syst- atic study of identification with quantized data is still in its early stage. This book presents new methodologies that utilize quantized information in system identification and explores their potential in extending control capabilities for systems with limited sensor information or networked systems. The book is an outgrowth of our recent research on quantized iden-—cation; it offers several salient features. From the viewpoint of targeted plants, it treats both linear and nonlinear systems, and both time-invariant and time-varying systems. In terms of noise types, it includes independent and dependent noises, shastic disturbances and deterministic bounded noises, and noises with unknown distribution functions. The key meth- ologies of the book combine empirical measures and information-theoretic approaches to cover convergence, convergence rate, estimator efficiency, - put design, threshold selection, and complexity analysis. We hope that it can shed new insights and perspectives for system identification.

Product Details

ISBN-13:	9780817649555
Publisher:	Birkhäuser Boston
Publication date:	05/25/2010
Series:	Systems & Control: Foundations & Applications
Edition description:	2010
Pages:	317
Product dimensions:	6.56(w) x 9.60(h) x 1.00(d)

Table of Contents

Preface xiii

Conventions xv

Glossary of Symbols xvii

Part I Overview 1

1 Introduction 3

1.1 Motivating Examples 4

1.2 System Identification with Quantized Observations 7

1.3 Outline of the Book 8

2 System Settings 13

2.1 Basic Systems 14

2.2 Quantized Output Observations 16

2.3 Inputs 17

2.4 System Configurations 18

2.4.1 Filtering and Feedback Configurations 19

2.4.2 Systems with Communication Channels 19

2.5 Uncertainties 20

2.5.1 System Uncertainties: Unmodeled Dynamics 20

2.5.2 System Uncertainties: Function Mismatch 21

2.5.3 Sensor Bias and Drifts 21

2.5.4 Noise 21

2.5.5 Unknown Noise Characteristics 22

2.5.6 Communication Channel Uncertainties 22

2.6 Notes 22

Part II Stochastic Methods for Linear Systems 23

3 Empirical-Measure-Based Identification 25

3.1 An Overview of Empirical-Measure-Based Identification 26

3.2 Empirical Measures and Identification Algorithms 29

3.3 Strong Convergence 32

3.4 Asymptotic Distributions 34

3.5 Mean-Square Convergence 37

3.6 Convergence under Dependent Noise 41

3.7 Proofs of Two Propositions 43

3.8 Notes 46

4 Estimation Error Bounds: Including Unmodeled Dynamics 49

4.1 Worst-Case Probabilistic Errors and Time Complexity 50

4.2 Upper Bounds on Estimation Errors and Time Complexity 50

4.3 Lower Bounds on Estimation Errors 53

4.4 Notes 56

5 Rational Systems 59

5.1 Preliminaries 59

5.2 Estimation of x_k 60

5.3 Estimation of Parameter θ 62

5.3.1 Parameter Identifiability 62

5.3.2 Identification Algorithms and Convergence Analysis 65

5.4 Notes 66

6 Quantized Identification and Asymptotic Efficiency 67

6.1 Basic Algorithms and Convergence 68

6.2 Quasi-Convex Combination Estimators (QCCE) 70

6.3 Alternative Covariance Expressions of Optimal QCCEs 72

6.4 Cramér-Rao Lower Bounds and Asymptotic Efficiency of the Optimal QCCE 75

6.5 Notes 79

7 Input Design for Identification in Connected Systems 81

7.1 Invariance of Input Periodicity and Rank in Open- and Closed-Loop Configurations 82

7.2 Periodic Dithers 83

7.3 Sufficient Richness Conditions under Input Noise 85

7.4 Actuator Noise 88

7.5 Notes 91

8 Identification of Sensor Thresholds and Noise Distribution Functions 95

8.1 Identification of Unknown Thresholds 95

8.1.1 Sufficient Richness Conditions 96

8.1.2 Recursive Algorithms 99

8.2 Parameterized Distribution Functions 99

8.3 Joint Identification Problems 101

8.4 Richness Conditions for Joint Identification 101

8.5 Algorithms for Identifying System Parameters and Distribution Functions 103

8.6 Convergence Analysis 105

8.7 Recursive Algorithms 106

8.7.1 Recursive Schemes 107

8.7.2 Asymptotic Properties of Recursive Algorithm (8.14) 108

8.8 Algorithm Flowcharts 111

8.9 Illustrative Examples 113

8.10 Notes 115

Part III Deterministic Methods for Linear Systems 117

9 Worst-Case Identification 119

9.1 Worst-Case Uncertainty Measures 120

9.2 Lower Bounds on Identification Errors and Time Complexity 121

9.3 Upper Bounds on Time Complexity 124

9.4 Identification of Gains 127

9.5 Identification Using Combined Deterministic and Stochastic Methods 135

9.5.1 Identifiability Conditions and Properties under Deterministic and Stochastic Frameworks 136

9.5.2 Combined Deterministic and Stochastic Identification Methods 139

9.5.3 Optimal Input Design and Convergence Speed under Typical Distributions 141

9.6 Notes 145

10 Worst-Case Identification Using Quantized Observations 149

10.1 Worst-Case Identification with Quantized Observations 150

10.2 Input Design for Parameter Decoupling 151

10.3 Identification of Single-Parameter Systems 153

10.3.1 General Quantization 154

10.3.2 Uniform Quantization 159

10.4 Time Complexity 163

10.5 Examples 165

10.6 Notes 168

Part IV Identification of Nonlinear and Switching Systems 171

11 Identification of Wiener Systems 173

11.1 Wiener Systems 174

11.2 Basic Input Design and Core Identification Problems 175

11.3 Properties of Inputs and Systems 177

11.4 Identification Algorithms 179

11.5 Asymptotic Efficiency of the Core Identification Algorithms 184

11.6 Recursive Algorithms and Convergence 188

11.7 Examples 190

11.8 Notes 194

12 Identification of Hammerstein Systems 197

12.1 Problem Formulation 198

12.2 Input Design and Strong-Full-Rank Signals 199

12.3 Estimates of ζ with Individual Thresholds 202

12.4 Quasi-Convex Combination Estimators of ζ 204

12.5 Estimation of System Parameters 212

12.6 Examples 218

12.7 Notes 222

13 Systems with Markovian Parameters 225

13.1 Markov Switching Systems with Binary Observations 227

13.2 Wonham-Type Filters 227

13.3 Tracking: Mean-Square Criteria 229

13.4 Tracking Infrequently Switching Systems: MAP Methods 237

13.5 Tracking Fast-Switching Systems 242

13.5.1 Long-Run Average Behavior 243

13.5.2 Empirical Measure-Based Estimators 245

13.5.3 Estimation Errors on Empirical Measures: Upper and Lower Bounds 249

13.6 Notes 252

Part V Complexity Analysis 253

14 Complexities, Threshold Selection, Adaptation 255

14.1 Space and Time Complexities 256

14.2 Binary Sensor Threshold Selection and Input Design 259

14.3 Worst-Case Optimal Threshold Design 261

14.4 Threshold Adaptation 264

14.5 Quantized Sensors and Optimal Resource Allocation 267

14.6 Discussions on Space and Time Complexity 271

14.7 Notes 272

15 Impact of Communication Channels 275

15.1 Identification with Communication Channels 276

15.2 Monotonicity of Fisher Information 277

15.3 Fisher Information Ratio of Communication Channels 278

15.4 Vector-Valued Parameters 280

15.5 Relationship to Shannon's Mutual Information 282

15.6 Tradeoff between Time Information and Space Information 283

15.7 Interconnections of Communication Channels 284

15.8 Notes 285

A Background Materials 287

A.1 Martingales 287

A.2 Markov Chains 290

A.3 Weak Convergence 299

A.4 Miscellany 302

References 305

Index 315

Editorial Reviews

From the reviews:

“The central idea in this book is to provide a comprehensive treatment of both theory and algorithms needed for parameter identification of systems with quantized observations. … the book conveys a clear and very complete overview of recent exciting developments in the area of identification with quantized observations. It is meant as a ‘state-of-the-art’ book … . All this makes the book an extremely valuable resource for researchers and engineers interested in modern system identification.” (Dariusz Uciński, Mathematical Reviews, Issue 2011 i)

From the Publisher