Recursive Identification and Parameter Estimation describes a recursive approach to solving system identification and parameter estimation problems arising from diverse areas. Supplying rigorous theoretical analysis, it presents the material and proposed algorithms in a manner that makes it easy to understand—providing readers with the modeling and identification skills required for successful theoretical research and effective application.
The book begins by introducing the basic concepts of probability theory, including martingales, martingale difference sequences, Markov chains, mixing processes, and stationary processes. Next, it discusses the root-seeking problem for functions, starting with the classic RM algorithm, but with attention mainly paid to the stochastic approximation algorithms with expanding truncations (SAAWET) which serves as the basic tool for recursively solving the problems addressed in the book.
The book not only identifies the results of system identification and parameter estimation, but also demonstrates how to apply the proposed approaches for addressing problems in a range of areas, including:
Identification of ARMAX systems without imposing restrictive conditions
Identification of typical nonlinear systems
Optimal adaptive tracking
Consensus of multi-agents systems
Principal component analysis
Distributed randomized PageRank computation
This book recursively identifies autoregressive and moving average with exogenous input (ARMAX) and discusses the identification of non-linear systems. It concludes by addressing the problems arising from different areas that are solved by SAAWET. Demonstrating how to apply the proposed approaches to solve problems across a range of areas, the book is suitable for students, researchers, and engineers working in systems and control, signal processing, communication, and mathematical statistics.

Preface xi

Acknowledgments xv

About the Authors xvii

1 Dependent Random Vectors 1

1.1 Some Concepts of Probability Theory 2

1.2 Independent Random Variables, Martingales, and Martingale Difference Sequences 8

1.3 Markov Chains with State Space (Rm,Bm) 15

1.4 Mixing Random Processes 29

1.5 Stationary Processes 33

1.6 Notes and References 40

2 Recursive Parameter Estimation 43

2.1 Parameter Estimation as Root-Seeking for Functions 44

2.2 Classical Stochastic Approximation Method: RM Algorithm 46

2.3 Stochastic Approximation Algorithm with Expanding Truncations 51

2.4 SAAWET with Nonadditive Noise 63

2.5 Linear Regression Functions 71

2.6 Convergence Rate of SAAWET 77

2.7 Notes and References 80

3 Recursive Identification for ARMAX Systems 81

3.1 LS and ELS for Linear Systems 82

3.2 Estimation Errors of LS/ELS 86

3.3 Hankel Matrices Associated with ARMA 93

3.4 Coefficient Identification of ARMAX by SAAWET 121

3.5 Order Estimation of ARMAX 142

3.6 Multivariate Linear EIV Systems 158

3.7 Notes and References 164

4 Recursive Identification for Nonlinear Systems 165

4.1 Recursive Identification of Hammerstein Systems 166

4.2 Recursive Identification of Wiener Systems 180

4.3 Recursive Identification of Wiener–Hammerstein Systems 195

4.4 Recursive Identification of EIV Hammerstein Systems 230

4.5 Recursive Identification of EIV Wiener Systems 253

4.6 Recursive Identification of Nonlinear ARX Systems 273

4.7 Notes and References 287

5 Other Problems Reducible to Parameter Estimation 289

5.1 Principal Component Analysis 289

5.2 Consensus of Networked Agents 316

5.3 Adaptive Regulation for Hammerstein and Wiener Systems 324

5.4 Convergence of Distributed Randomized PageRank Algorithms 337

5.5 Notes and References 352

Appendix A: Proof of Some Theorems in Chapter 1 353

Appendix B: Nonnegative Matrices 381

References 397

Index 407

Wenxiao Zhao received his BSc degree from the Department of Mathematics, Shandong University, China in 2003 and the PhD degree from the Institute of Systems Science, AMSS, the Chinese Academy of Sciences (CAS) in 2008. After this he was a postdoctoral student at the Department of Automation, Tsinghua University.PA\During this period he visited the University of Western Sydney, Australia, for nine months. Then, Dr. Zhao joined the Institute of Systems Sciences, CAS in 2010. He now is with the Key Laboratory of Systems and Control, CAS as an associate professor. His research interests are in system identification, adaptive control, and system biology. He serves as the general secretary of the IEEE Control Systems Beijing Chapter and an associate editor of the Journal of Systems Science and Mathematical Sciences.

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