Lieve Lauwers Some practical applications of the best linear approximation in nonlinear block ...

207 Pages·2011·3.27 MB·English

by  Lieve Lauwers

FACULTY OF ENGINEERING
Department of Fundamental Electricity and Instrumentation
Pleinlaan 2 - 1050 Brussels
Some practical applications of
the best linear approximation in nonlinear
block-oriented modelling
Thesis submitted in fulfilment of the requirements for the degree of
Doctor in Engineering (Doctor in de Ingenieurswetenschappen) by
Lieve Lauwers
Chair: Prof. Dr. ir. Gert Desmet (Vrije Universiteit Brussel)
Vice chair: Prof. Dr. ir. Herman Terryn (Vrije Universiteit Brussel)
Secretary: Dr. Kurt Barbé (Vrije Universiteit Brussel)
Adviser: Prof. Dr. ir. Johan Schoukens (Vrije Universiteit Brussel)
Jury: Em. Prof. Dr. ir. Keith Godfrey (University of Warwick)
Prof. Dr. ir. Joos Vandewalle (KULeuven)
Prof. Dr. Steve Vanlanduit (Vrije Universiteit Brussel)
© 2011 (cid:47)(cid:76)(cid:72)(cid:89)(cid:72)(cid:3)(cid:47)(cid:68)(cid:88)(cid:90)(cid:72)(cid:85)(cid:86)

2011 Uitgeverij University Press
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ISBN 978-94-9069-56(cid:28)(cid:16)(cid:28)
Legal Deposit D/2011/11.161/01(cid:28)
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written permission of the author.
To the ones I love...

Acknowledgements
I would like to thank everyone that has contributed to this work in one way or
another.
I thank Johan Schoukens for giving me the opportunity to pursue my PhD at
the ELEC department. I wish to thank all my colleagues for the pleasant work
environment, especially my office mates: Koen, Mohamed and Charles. I know I
was not always easy to share the office with...
Aspecialwordofthanksgoestomythreemusketeers: Joke, WendyandKurt.
Thanksalotforbelievinginmeandforgivingmethestrengthtogoon. Ienjoyed
ourrelaxingtalksandthefuntimewespentatthedepartmentandatconferences.
Mostofall,Iamgratefultomyfamilyfortheirendlessloveandsupport. Thank
you for encouraging me all the way! I am who I am, because of you!
I fall short of words when it comes to thank my soulmate, Joke. Thank you
for loving me for who I am, for your patience and understanding. Without you, I
could not have done it!
Lieve Lauwers
May 2011

Contents
Operators and Notational Conventions xi
Symbols xiii
Abbreviations xv
1 Introduction 1
1.1 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Linear versus Nonlinear Systems . . . . . . . . . . . . . . . . . . 2
1.3 System Identification Process . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Collect data . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Select a model . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Fit the model to the data . . . . . . . . . . . . . . . . . . 6
1.3.4 Validate the model . . . . . . . . . . . . . . . . . . . . . 7
1.3.5 Iterative process . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.6 Some important questions . . . . . . . . . . . . . . . . . 7
1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
I Modelling using the Best Linear Approximation 15
2 Some Block-oriented Systems 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Basic Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Linear time-invariant systems . . . . . . . . . . . . . . . . 18
2.2.2 Static and dynamic nonlinearities . . . . . . . . . . . . . . 19
2.3 Specific Block Structures . . . . . . . . . . . . . . . . . . . . . . 20
v
Contents
2.3.1 Wiener systems . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Hammerstein systems . . . . . . . . . . . . . . . . . . . . 21
2.3.3 Wiener-Hammerstein systems . . . . . . . . . . . . . . . . 22
2.3.4 Hammerstein-Wiener systems . . . . . . . . . . . . . . . . 22
2.3.5 Nonlinear feedback systems . . . . . . . . . . . . . . . . . 23
2.3.6 Nonlinear parallel systems . . . . . . . . . . . . . . . . . . 24
2.4 Parallel block-oriented models in a broader context . . . . . . . . 25
2.5 Identification Issues . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1 How to select a model structure? . . . . . . . . . . . . . . 28
2.5.2 How to initialise the model? . . . . . . . . . . . . . . . . 29
3 The Best Linear Approximation 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Approximation Criterion . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Defining the Best Linear Approximation . . . . . . . . . . . . . . 33
3.4 Class of Excitation Signals . . . . . . . . . . . . . . . . . . . . . 36
3.4.1 Gaussian noise . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4.2 Random phase multisine . . . . . . . . . . . . . . . . . . 37
3.5 Properties of the Best Linear Approximation . . . . . . . . . . . . 39
3.5.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.2 Representation . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Estimation of the Best Linear Approximation. . . . . . . . . . . . 42
3.6.1 Periodic data . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6.2 Nonperiodic data . . . . . . . . . . . . . . . . . . . . . . 45
3.6.3 Summary of notation . . . . . . . . . . . . . . . . . . . . 46
4 Model Structure Selection Method 47
4.1 Existing Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Model Structure Selection Method . . . . . . . . . . . . . . . . . 48
4.2.1 Two-step nonparametric approach . . . . . . . . . . . . . 49
4.2.2 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.3 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Behaviour of the Best Linear Approximation . . . . . . . . . . . . 50
4.3.1 Simulation setup. . . . . . . . . . . . . . . . . . . . . . . 50
4.3.2 Static nonlinearity . . . . . . . . . . . . . . . . . . . . . . 51
4.3.3 Dynamic nonlinearity: NFIR system . . . . . . . . . . . . 51
4.3.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.5 Wiener-Hammerstein system . . . . . . . . . . . . . . . . 59
4.3.6 WH-NFIR system . . . . . . . . . . . . . . . . . . . . . . 61
4.3.7 Nonlinear feedback system . . . . . . . . . . . . . . . . . 65
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Contents
4.3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Practical Applications . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . 69
4.5.2 Case study 1: Silverbox . . . . . . . . . . . . . . . . . . . 69
4.5.3 Case study 2: Crystal Detector . . . . . . . . . . . . . . . 72
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Appendix 4.A Calculation of the Uncertainty Bounds . . . . . . . . . . 76
5 On the Initialisation of Wiener-Hammerstein Models 79
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Initialisation Algorithm . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.2 Decomposing the WH structure . . . . . . . . . . . . . . 81
5.2.3 Extracting the poles and zeros from GBLA . . . . . . . . 82
5.2.4 Constructing the basis functions . . . . . . . . . . . . . . 83
5.2.5 Solving a problem linear-in-the-parameters . . . . . . . . . 84
5.2.6 Composing the initial estimates . . . . . . . . . . . . . . . 86
5.2.7 Numerical issues . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.9 Derived approach . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Nonlinear Optimisation . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Practical Application . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5.1 Description of the DUT . . . . . . . . . . . . . . . . . . . 97
5.5.2 Description of the data . . . . . . . . . . . . . . . . . . . 97
5.5.3 Best linear approximation . . . . . . . . . . . . . . . . . . 98
5.5.4 Initial estimates . . . . . . . . . . . . . . . . . . . . . . . 99
5.5.5 Nonlinear optimisation . . . . . . . . . . . . . . . . . . . 102
5.5.6 Benchmark results . . . . . . . . . . . . . . . . . . . . . . 103
5.5.7 Comparison with other approaches . . . . . . . . . . . . . 106
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Appendix 5.A Study of the Delay Terms . . . . . . . . . . . . . . . . . 110
Appendix 5.B Model Equivalence Proof . . . . . . . . . . . . . . . . . 112
Appendix 5.C Hinge Functions . . . . . . . . . . . . . . . . . . . . . . 113
II Modelling using Prior Knowledge 115
6 Parameter Estimation of the Rice Distribution 117
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Contents
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Bayesian Framework . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.1 Prior knowledge . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.2 Methodology. . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Rice Estimation Problem . . . . . . . . . . . . . . . . . . . . . . 120
6.3.1 Rice distribution . . . . . . . . . . . . . . . . . . . . . . . 120
6.3.2 Problem statement . . . . . . . . . . . . . . . . . . . . . 121
6.3.3 Existing approaches . . . . . . . . . . . . . . . . . . . . . 122
6.4 Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4.1 Prior density . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4.2 Maximum A Posteriori estimator . . . . . . . . . . . . . . 130
6.5 Comparison of Estimators . . . . . . . . . . . . . . . . . . . . . . 132
6.5.1 Dependency of starting values . . . . . . . . . . . . . . . 132
6.5.2 Performance as a function of SNR . . . . . . . . . . . . . 133
6.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.6 Study of ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.6.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 138
6.6.2 Parameter uncertainty . . . . . . . . . . . . . . . . . . . . 140
6.6.3 Sensitivity analysis. . . . . . . . . . . . . . . . . . . . . . 142
6.6.4 User guideline . . . . . . . . . . . . . . . . . . . . . . . . 143
6.7 Practical Application to fMRI Data . . . . . . . . . . . . . . . . . 143
6.7.1 Background to fMRI . . . . . . . . . . . . . . . . . . . . 143
6.7.2 Processing fMRI data . . . . . . . . . . . . . . . . . . . . 145
6.7.3 Description of the data . . . . . . . . . . . . . . . . . . . 147
6.7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Appendix 6.A Property on the Prior Density . . . . . . . . . . . . . . . 152
Appendix 6.B Existence of a Normalising Constant . . . . . . . . . . . 153
Appendix 6.C Uniqueness of the MAP Solution . . . . . . . . . . . . . 157
Appendix 6.D Explicit Expressions for the Jacobian . . . . . . . . . . . 165
7 Conclusions 167
7.1 Overall Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.2.1 Extending the model structure selection method . . . . . . 169
7.2.2 Automated distinction of model structures . . . . . . . . . 169
7.2.3 Reducing the number of model parameters. . . . . . . . . 169
7.2.4 Designing optimal statistical tests . . . . . . . . . . . . . 170
7.2.5 Modelling the dynamics in fMRI signals . . . . . . . . . . 170
viii