Nonlinear System Identification – NARMAX Methods in the Time, Frequency and Spatio–Temporal Domains

Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains describes a comprehensive framework for the identification and analysis of nonlinear dynamic systems in the time, frequency, and spatio-temporal domains. This book is written with an emphasis on making the algorithms accessible so that they can be applied and used in practice. Includes coverage of: * The NARMAX (nonlinear autoregressive moving average with exogenous inputs) model * The orthogonal least squares algorithm that allows models to be built term by term where the error reduction ratio reveals the percentage contribution of each model term * Statistical and qualitative model validation methods that can be applied to any model class * Generalised frequency response functions which provide significant insight into nonlinear behaviours * A completely new class of filters that can move, split, spread, and focus energy * The response spectrum map and the study of sub harmonic and severely nonlinear systems * Algorithms that can track rapid time variation in both linear and nonlinear systems * The important class of spatio-temporal systems that evolve over both space and time * Many case study examples from modelling space weather, through identification of a model of the visual processing system of fruit flies, to tracking causality in EEG data are all included to demonstrate how easily the methods can be applied in practice and to show the insight that the algorithms reveal even for complex systems NARMAX algorithms provide a fundamentally different approach to nonlinear system identification and signal processing for nonlinear systems. NARMAX methods provide models that are transparent, which can easily be analysed, and which can be used to solve real problems. This book is intended for graduates, postgraduates and researchers in the sciences and engineering, and also for users from other fields who have collected data and who wish to identify models to help to understand the dynamics of their systems.

Table of Contents:
Preface xv 1 Introduction 1 1.1 Introduction to System Identification 1 1.2 Linear System Identification 3 1.3 Nonlinear System Identification 5 1.4 NARMAX Methods 7 1.5 The NARMAX Philosophy 8 1.6 What is System Identification For? 9 1.7 Frequency Response of Nonlinear Systems 11 1.8 Continuous-Time, Severely Nonlinear, and Time-Varying Models and Systems 12 1.9 Spatio-temporal Systems 13 1.10 Using Nonlinear System Identification in Practice and Case Study Examples 13 References 14 2 Models for Linear and Nonlinear Systems 17 2.1 Introduction 17 2.2 Linear Models 18 2.3 Piecewise Linear Models 22 2.4 Volterra Series Models 30 2.5 Block-Structured Models 31 2.6 NARMAX Models 33 2.7 Generalised Additive Models 40 2.8 Neural Networks 41 2.9 Wavelet Models 45 2.10 State-Space Models 48 2.11 Extensions to the MIMO Case 49 2.12 Noise Modelling 49 2.13 Spatio-temporal Models 52 References 53 3 Model Structure Detection and Parameter Estimation 61 3.1 Introduction 61 3.2 The Orthogonal Least Squares Estimator and the Error Reduction Ratio 64 Representation 65 3.3 The Forward Regression OLS Algorithm 70 3.4 Term and Variable Selection 79 3.5 OLS and Sum of Error Reduction Ratios 80 3.6 Noise Model Identification 84 3.7 An Example of Variable and Term Selection for a Real Data Set 87 3.8 ERR is Not Affected by Noise 94 3.9 Common Structured Models to Accommodate Different Parameters 95 3.10 Model Parameters as a Function of Another Variable 98 3.11 OLS and Model Reduction 100 3.12 Recursive Versions of OLS 102 References 102 4 Feature Selection and Ranking 105 4.1 Introduction 105 4.2 Feature Selection and Feature Extraction 106 4.3 Principal Components Analysis 107 4.4 A Forward Orthogonal Search Algorithm 108 4.5 A Basis Ranking Algorithm Based on PCA 113 References 117 5 Model Validation 119 5.1 Introduction 119 5.2 Detection of Nonlinearity 121 5.3 Estimation and Test Data Sets 123 5.4 Model Predictions 124 5.5 Statistical Validation 127 5.6 Term Clustering 135 5.7 Qualitative Validation of Nonlinear Dynamic Models 137 References 145 6 The Identification and Analysis of Nonlinear Systems in the Frequency Domain 149 6.1 Introduction 149 6.2 Generalised Frequency Response Functions 151 6.3 Output Frequencies of Nonlinear Systems 184 6.4 Nonlinear Output Frequency Response Functions 191 6.5 Output Frequency Response Function of Nonlinear Systems 202 References 213 7 Design of Nonlinear Systems in the Frequency Domain Energy Transfer Filters and Nonlinear Damping 217 7.1 Introduction 217 7.2 Energy Transfer Filters 218 7.3 Energy Focus Filters 240 7.4 OFRF-Based Approach for the Design of Nonlinear Systems in the Frequency Domain 249 References 259 8 Neural Networks for Nonlinear System Identification 261 8.1 Introduction 261 8.2 The Multi-layered Perceptron 263 8.3 Radial Basis Function Networks 264 8.4 Wavelet Networks 270 8.5 Multi-resolution Wavelet Models and Networks 277 References 284 9 Severely Nonlinear Systems 289 9.1 Introduction 289 9.2 Wavelet NARMAX Models 291 9.3 Systems that Exhibit Sub-harmonics and Chaos 301 9.4 The Response Spectrum Map 305 9.5 A Modelling Framework for Sub-harmonic and Severely Nonlinear Systems 313 9.6 Frequency Response Functions for Sub-harmonic Systems 320 9.7 Analysis of Sub-harmonic Systems and the Cascade to Chaos 326 References 334 10 Identification of Continuous-Time Nonlinear Models 337 10.1 Introduction 337 10.2 The Kernel Invariance Method 338 10.3 Using the GFRFs to Reconstruct Nonlinear Integro-differential Equation Models Without Differentiation 352 References 367 11 Time-Varying and Nonlinear System Identification 371 11.1 Introduction 371 11.2 Adaptive Parameter Estimation Algorithms 372 11.3 Tracking Rapid Parameter Variations Using Wavelets 376 11.4 Time-Dependent Spectral Characterisation 378 11.5 Nonlinear Time-Varying Model Estimation 380 11.6 Mapping and Tracking in the Frequency Domain 381 11.7 A Sliding Window Approach 388 References 389 12 Identification of Cellular Automata and N -State Models of Spatio-temporal Systems 391 12.1 Introduction 391 12.2 Cellular Automata 393 12.3 Identification of Cellular Automata 402 12.4 N -State Systems 414 References 427 13 Identification of Coupled Map Lattice and Partial Differential Equations of Spatio-temporal Systems 431 13.1 Introduction 431 13.2 Spatio-temporal Patterns and Continuous-State Models 432 13.3 Identification of Coupled Map Lattice Models 437 13.4 Identification of Partial Differential Equation Models 458 13.5 Nonlinear Frequency Response Functions for Spatio-temporal Systems 466 References 471 14 Case Studies 473 14.1 Introduction 473 14.2 Practical System Identification 474 14.3 Characterisation of Robot Behaviour 478 14.4 System Identification for Space Weather and the Magnetosphere 484 14.5 Detecting and Tracking Iceberg Calving in Greenland 493 14.6 Detecting and Tracking Time-Varying Causality for EEG Data 498 14.7 The Identification and Analysis of Fly Photoreceptors 505 14.8 Real-Time Diffuse Optical Tomography Using RBF Reduced-Order Models of the Propagation of Light for Monitoring Brain Haemodynamics 514 14.9 Identification of Hysteresis Effects in Metal Rubber Damping Devices 522 14.10 Identification of the Belousov Zhabotinsky Reaction 528 14.11 Dynamic Modelling of Synthetic Bioparts 534 14.12 Forecasting High Tides in the Venice Lagoon 539 References 543 Index

About the Author :
Stephen A. Billings, University of Sheffield, UK Stephen A. Billings is Professor of Signal Processing and Complex Systems, and Director of the Signal Processing and Complex Systems Research Group, in the Department of Automatic Control and Systems Engineering at the University of Sheffield, He is counted as "highly cited" by the ISI Web of Knowledge with 250 publications to his name.