Preface xiii

Acknowledgments xv

Abbreviations xvii

**1 Identification 1**

1.1 Introduction 1

1.2 Illustration of Some Important Aspects of System Identification 2

Exercise 1 .a (Least squares estimation of the value of a resistor) 2

Exercise 1 .b (Analysis of the standard deviation) 3

Exercise 2 (Study of the asymptotic distribution of an estimate) 5

Exercise 3 (Impact of noise on the regressor (input) measurements) 6

Exercise 4 (Importance of the choice of the independent variable or input) 7

Exercise 5.a (combining measurements with a varying SNR: Weighted least squares estimation) 8

Exercise 5.b (Weighted least squares estimation: A study of the variance) 9

Exercise 6 (Least squares estimation of models that are linear in the parameters) 11

Exercise 7 (Characterizing a 2-dimensional parameter estimate) 12

1.3 Maximum Likelihood Estimation for Gaussian and Laplace Distributed Noise 14

Exercise 8 (Dependence of the optimal cost function on the distribution of the disturbing noise) 14

1.4 Identification for Skew Distributions with Outliers 16

Exercise 9 (Identification in the presence of outliers) 16

1.5 Selection of the Model Complexity 18

Exercise 10 (Influence of the number of parameters on the model uncertainty) 18

Exercise 11 (Model selection using the AIC criterion) 20

1.6 Noise on Input and Output Measurements: The IV Method and the EIV Method 22

Exercise 12 (Noise on input and output: The instrumental variables method applied on the resistor estimate) 23

Exercise 13 (Noise on input and output: the errors-in-variables method) 25

**2 Generation and Analysis of Excitation Signals 29**

2.1 Introduction 29

2.2 The Discrete Fourier Transform (DFT) 30

Exercise 14 (Discretization in time: Choice of the sampling frequency: ALIAS) 31

Exercise 15 (Windowing: Study of the leakage effect and the frequency resolution) 31

2.3 Generation and Analysis of Multisines and Other Periodic Signals 33

Exercise 16 (Generate a sine wave, noninteger number of periods measured) 34

Exercise 17 (Generate a sine wave, integer number of periods measured) 34

Exercise 18 (Generate a sine wave, doubled measurement time) 35

Exercise 19.a (Generate a sine wave using the MATLAB IFFT instruction) 37

Exercise 19.b (Generate a sine wave using the MATLAB IFFT instruction, defining only the first half of the spectrum) 37

Exercise 20 (Generation of a multisine with flat amplitude spectrum) 38

Exercise 21 (The swept sine signal) 39

Exercise 22.a (Spectral analysis of a multisine signal, leakage present) 40

Exercise 22.b (Spectral analysis of a multisine signal, no leakage present) 40

2.4 Generation of Optimized Periodic Signals 42

Exercise 23 (Generation of a multisine with a reduced crest factor using random phase generation) 42

Exercise 24 (Generation of a multisine with a minimal crest factor using a crest factor minimization algorithm) 42

Exercise 25 (Generation of a maximum length binary sequence) 45

Exercise 26 (Tuning the parameters of a maximum length binary sequence) 46

2.5 Generating Signals Using The Frequency Domain Identification Toolbox (FDIDENT) 46

Exercise 27 (Generation of excitation signals using the FDIDENT toolbox) 47

2.6 Generation of Random Signals 48

Exercise 28 (Repeated realizations of a white random noise excitation with fixed length) 48

Exercise 29 (Repeated realizations of a white random noise excitation with increasing length) 49

Exercise 30 (Smoothing the amplitude spectrum of a random excitation) 49

Exercise 31 (Generation of random noise excitations with a user-imposed power spectrum) 50

Exercise 32 (Amplitude distribution of filtered noise) 51

2.7 Differentiation, Integration, Averaging, and Filtering of Periodic Signals 52

Exercise 33 (Exploiting the periodic nature of signals: Differentiation, integration, +averaging, and filtering) 52

**3 FRF Measurements 55**

3.1 Introduction 55

3.2 Definition of the FRF 56

3.3 FRF Measurements without Disturbing Noise 57

Exercise 34 (Impulse response function measurements) 57

Exercise 35 (Study of the sine response of a linear system: transients and steady-state) 58

Exercise 36 (Study of a multisine response of a linear system: transients and steady-state) 59

Exercise 37 (FRF measurement using a noise excitation and a rectangular window) 61

Exercise 38 (Revealing the nature of the leakage effect in FRF measurements) 61

Exercise 39 (FRF measurement using a noise excitation and a Hanning window) 64

Exercise 40 (FRF measurement using a noise excitation and a diff window) 65

Exercise 41 (FRF measurements using a burst excitation) 66

3.4 FRF Measurements in the Presence of Disturbing Output Noise 68

Exercise 42 (Impulse response function measurements in the presence of output noise) 69

Exercise 43 (Measurement of the FRF using a random noise sequence and a random phase multisine in the presence of output noise) 70

Exercise 44 (Analysis of the noise errors on FRF measurements) 71

Exercise 45 (Impact of the block (period) length on the uncertainty) 73

3.5 FRF Measurements in the Presence of Input and Output Noise 75

Exercise 46 (FRF measurement in the presence of input/output disturbances using a multisine excitation) 75

Exercise 47 (Measuring the FRF in the presence of input and output noise: Analysis of the errors) 75

Exercise 48 (Measuring the FRF in the presence of input and output noise: Impact of the block (period) length on the uncertainty) 76

3.6 FRF Measurements of Systems Captured in a Feedback Loop 78

Exercise 49 (Direct measurement of the FRF under feedback conditions) 78

Exercise 50 (The indirect method) 80

3.7 FRF Measurements Using Advanced Signal Processing Techniques: The LPM 82

Exercise 51 (The local polynomial method) 82

Exercise 52 (Estimation of the power spectrum of the disturbing noise) 84

3.8 Frequency Response Matrix Measurements for MIMO Systems 85

Exercise 53 (Measuring the FRM using multisine excitations) 85

Exercise 54 (Measuring the FRM using noise excitations) 86

Exercise 55 (Estimate the variance of the measured FRM) 88

Exercise 56 (Comparison of the actual and theoretical variance of the estimated FRM) 88

Exercise 57 (Measuring the FRM using noise excitations and a Hanning window) 89

**4 Identification of Linear Dynamic Systems 91**

4.1 Introduction 91

4.2 Identification Methods that Are Linear-in-the-Parameters. The Noiseless Setup 93

Exercise 58 (Identification in the time domain) 94

Exercise 59 (Identification in the frequency domain) 96

Exercise 60 (Numerical conditioning) 97

Exercise 61 (Simulation and one-step-ahead prediction) 99

Exercise 62 (Identify a too-simple model) 100

Exercise 63 (Sensitivity of the simulation and prediction error to model errors) 101

Exercise 64 (Shaping the model errors in the time domain: Prefiltering) 102

Exercise 65 (Shaping the model errors in the frequency domain: frequency weighting) 102

4.3 Time domain Identification using parametric noise models 104

Exercise 66 (One-step-ahead prediction of a noise sequence) 105

Exercise 67 (Identification in the time domain using parametric noise models) 108

Exercise 68 (Identification Under Feedback Conditions Using Time Domain Methods) 109

Exercise 69 (Generating uncertainty bounds for estimated models) 111

Exercise 70 (Study of the behavior of the BJ model in combination with prefiltering) 113

4.4 Identification Using Nonparametric Noise Models and Periodic Excitations 115

Exercise 71 (Identification in the frequency domain using nonparametric noise models) 117

Exercise 72 (Emphasizing a frequency band) 119

Exercise 73 (Comparison of the time and frequency domain identification under feedback) 120

4.5 Frequency Domain Identification Using Nonparametric Noise Models and Random Excitations 122

Exercise 74 (Identification in the frequency domain using nonparametric noise models and a random excitation) 122

4.6 Time Domain Identification Using the System Identification Toolbox 123

Exercise 75 (Using the time domain identification toolbox) 124

4.7 Frequency Domain Identification Using the Toolbox FDIDENT 129

Exercise 76 (Using the frequency domain identification toolbox FDIDENT) 129

**5 Best Linear Approximation of Nonlinear Systems 137**

5.1 Response of a nonlinear system to a periodic input 137

Exercise 77.a (Single sine response of a static nonlinear system) 138

Exercise 77.b (Multisine response of a static nonlinear system) 139

Exercise 78 (Uniform versus Pointwise Convergence) 142

Exercise 79.a (Normal operation, subharmonics, and chaos) 143

Exercise 79.b (Influence initial conditions) 146

Exercise 80 (Multisine response of a dynamic nonlinear system) 147

Exercise 81 (Detection, quantification, and classification of nonlinearities) 148

5.2 Best Linear Approximation of a Nonlinear System 150

Exercise 82 (Influence DC values signals on the linear approximation) 151

Exercise 83.a (Influence of rms value and pdf on the BLA) 152

Exercise 83.b (Influence of power spectrum coloring and pdf on the BLA) 154

Exercise 83.c (Influence of length of impulse response of signal filter on the BLA) 156

Exercise 84.a (Comparison of Gaussian noise and random phase multisine) 158

Exercise 84.b (Amplitude distribution of a random phase multisine) 160

Exercise 84.c (Influence of harmonic content multisine on BLA) 162

Exercise 85 (Influence of even and odd nonlinearities on BLA) 165

Exercise 86 (BLA of a cascade) 167

5.3 Predictive Power of The Best Linear Approximation 172

Exercise 87.a (Predictive power BLA — static NL system) 172

Exercise 87.b (Properties of output residuals — dynamic NL system) 174

Exercise 87.c (Predictive power of BLA — dynamic NL system) 178

**6 Measuring the Best Linear Approximation of a Nonlinear System 183**

6.1 Measuring the Best Linear Approximation 183

Exercise 88.a (Robust method for noisy FRF measurements) 186

Exercise 88.b (Robust method for noisy input/output measurements without reference signal) 190

Exercise 88.c (Robust method for noisy input/output measurements with reference signal) 195

Exercise 89.a (Design of baseband odd and full random phase multisines with random harmonic grid) 197

Exercise 89.b (Design of bandpass odd and full random phase multisines with random harmonic grid) 197

Exercise 89.c (Fast method for noisy input/output measurements — open loop example) 203

Exercise 89.d (Fast method for noisy input/output measurements — closed loop example) 207

Exercise 89.e (Bias on the estimated odd and even distortion levels) 211

Exercise 90 (Indirect method for measuring the best linear approximation) 215

Exercise 91 (Comparison robust and fast methods) 216

Exercise 92 (Confidence intervals for the BLA) 219

Exercise 93 (Prediction of the bias contribution in the BLA) 221

Exercise 94 (True underlying linear system) 222

6.2 Measuring the nonlinear distortions 224

Exercise 95 (Prediction of the nonlinear distortions using random harmonic grid multisines) 225

Exercise 96 (Pros and cons full-random and odd-random multisines) 230

6.3 Guidelines 233

6.4 Projects 233

**7 Identification of Parametric Models in the Presence of Nonlinear Distortions 239**

7.1 Introduction 239

7.2 Identification of the Best Linear Approximation Using Random Excitations 240

Exercise 97 (Parametric estimation of the best linear approximation) 240

7.3 Generation of Uncertainty Bounds? 243

Exercise 98 243

7.4 Identification of the best linear approximation using periodic excitations 245

Exercise 99 (Estimate a parametric model for the best linear approximation using the Fast Method) 246

Exercise 100 (Estimating a parametric model for the best linear approximation using the robust method) 251

7.5 Advises and conclusions 252

References 255

Subject Index 259

Reference Index 263