Model Identification and Data Analysis

SERGIO BITTANTI is Emeritus Professor of Model Identification and Data Analysis (MIDA) at the Politecnico di Milano, Milan, Italy, where his intense activity of research and teaching has attracted the attention of many young researchers. He started teaching the course of MIDA years ago, with just a few students. Today the course is offered in various sections with about one thousand students. He has organized a number of workshops and conferences, and has served as member of the Program Committee of more than 70 international congresses. He has for many years been associated with the National Research Council (CNR) of Italy and is a member of the Academy of Science and Literature of Milan (Istituto Lombardo – Accademia di Scienze e Lettere). He received many awards, in particular the title of Ambassador of the city of Milan and the medal of the President of the Italian Republic for the IFAC World Congress held in Milan in 2011 with a record number of attendees from 73 Countries. Website: http://home.deib.polimi.it/bittanti/

Introduction xi

Acknowledgments xv

1 Stationary Processes and Time Series 1

1.1 Introduction 1

1.2 The Prediction Problem 1

1.3 Random Variable 4

1.4 Random Vector 5

1.4.1 Covariance Coefficient 7

1.5 Stationary Process 9

1.6 White Process 11

1.7 MA Process 12

1.8 AR Process 16

1.8.1 Study of the AR(1) Process 16

1.9 Yule–Walker Equations 20

1.9.1 Yule–Walker Equations for the AR(1) Process 20

1.9.2 Yule–Walker Equations for the AR(2) and AR(n) Process 21

1.10 ARMA Process 23

1.11 Spectrum of a Stationary Process 24

1.11.1 Spectrum Properties 24

1.11.2 Spectral Diagram 25

1.11.3 Maximum Frequency in Discrete Time 25

1.11.4 White Noise Spectrum 25

1.11.5 Complex Spectrum 26

1.12 ARMA Model: Stability Test and Variance Computation 26

1.12.1 Ruzicka Stability Criterion 28

1.12.2 Variance of an ARMA Process 32

1.13 FundamentalTheorem of Spectral Analysis 35

1.14 Spectrum Drawing 38

1.15 Proof of the FundamentalTheorem of Spectral Analysis 43

1.16 Representations of a Stationary Process 45

2 Estimation of Process Characteristics 47

2.1 Introduction 47

2.2 General Properties of the Covariance Function 47

2.3 Covariance Function of ARMA Processes 49

2.4 Estimation of the Mean 50

2.5 Estimation of the Covariance Function 53

2.6 Estimation of the Spectrum 55

2.7 Whiteness Test 57

3 Prediction 61

3.1 Introduction 61

3.2 Fake Predictor 62

3.2.1 Practical Determination of the Fake Predictor 64

3.3 Spectral Factorization 66

3.4 Whitening Filter 70

3.5 Optimal Predictor from Data 71

3.6 Prediction of an ARMA Process 76

3.7 ARMAX Process 77

3.8 Prediction of an ARMAX Process 78

4 Model Identification 81

4.1 Introduction 81

4.2 Setting the Identification Problem 82

4.2.1 Learning from Maxwell 82

4.2.2 A General Identification Problem 84

4.3 Static Modeling 85

4.3.1 Learning from Gauss 85

4.3.2 Least Squares Made Simple 86

4.3.2.1 Trend Search 86

4.3.2.2 Seasonality Search 86

4.3.2.3 Linear Regression 87

4.3.3 Estimating the Expansion of the Universe 90

4.4 Dynamic Modeling 92

4.5 External RepresentationModels 92

4.5.1 Box and Jenkins Model 92

4.5.2 ARX and AR Models 93

4.5.3 ARMAX and ARMA Models 94

4.5.4 MultivariableModels 96

4.6 Internal RepresentationModels 96

4.7 The Model Identification Process 100

4.8 The Predictive Approach 101

4.9 Models in Predictive Form 102

4.9.1 Box and Jenkins Model 103

4.9.2 ARX and AR Models 103

4.9.3 ARMAX and ARMA Models 104

5 Identification of Input–Output Models 107

5.1 Introduction 107

5.2 Estimating AR and ARX Models: The Least Squares Method 107

5.3 Identifiability 110

5.3.1 The ̄R Matrix for the ARX(1, 1) Model 111

5.3.2 The ̄R Matrix for a General ARX Model 112

5.4 Estimating ARMA and ARMAX Models 115

5.4.1 Computing the Gradient and the Hessian from Data 117

5.5 Asymptotic Analysis 123

5.5.1 Data Generation SystemWithin the Class of Models 125

5.5.2 Data Generation System Outside the Class of Models 127

5.5.2.1 Simulation Trial 132

5.5.3 General Considerations on the Asymptotics of Predictive Identification 132

5.5.4 Estimating the Uncertainty in Parameter Estimation 132

5.5.4.1 Deduction of the Formula of the Estimation Covariance 134

5.6 Recursive Identification 138

5.6.1 Recursive Least Squares 138

5.6.2 Recursive Maximum Likelihood 143

5.6.3 Extended Least Squares 145

5.7 Robustness of IdentificationMethods 147

5.7.1 Prediction Error and Model Error 147

5.7.2 Frequency Domain Interpretation 148

5.7.3 Prefiltering 149

5.8 Parameter Tracking 149

6 Model Complexity Selection 155

6.1 Introduction 155

6.2 Cross-validation 157

6.3 FPE Criterion 157

6.3.1 FPE Concept 157

6.3.2 FPE Determination 158

6.4 AIC Criterion 160

6.4.1 AIC Versus FPE 161

6.5 MDL Criterion 161

6.5.1 MDL Versus AIC 162

6.6 Durbin–Levinson Algorithm 164

6.6.1 Yule–Walker Equations for Autoregressive Models of Orders 1 and 2 165

6.6.2 Durbin–Levinson Recursion: From AR(1) to AR(2) 166

6.6.3 Durbin–Levinson Recursion for Models of Any Order 169

6.6.4 Partial Covariance Function 171

7 Identification of State Space Models 173

7.1 Introduction 173

7.2 Hankel Matrix 175

7.3 Order Determination 176

7.4 Determination of Matrices G and H 177

7.5 Determination of Matrix F 178

7.6 Mid Summary: An Ideal Procedure 179

7.7 Order Determination with SVD 179

7.8 Reliable Identification of a State Space Model 181

8 Predictive Control 187

8.1 Introduction 187

8.2 Minimum Variance Control 188

8.2.1 Determination of the MV Control Law 190

8.2.2 Analysis of the MV Control System 192

8.2.2.1 Structure 193

8.2.2.2 Stability 193

8.3 Generalized Minimum Variance Control 196

8.3.1 Model Reference Control 198

8.3.2 Penalized Control Design 200

8.3.2.1 Choice A for Q(z) 201

8.3.2.2 Choice B for Q(z) 203

8.4 Model-Based Predictive Control 204

8.5 Data-Driven Control Synthesis 205

9 Kalman Filtering and Prediction 209

9.1 Introduction 209

9.2 Kalman Approach to Prediction and Filtering Problems 210

9.3 The Bayes Estimation Problem 212

9.3.1 Bayes Problem – Scalar Case 213

9.3.2 Bayes Problem – Vector Case 215

9.3.3 Recursive Bayes Formula – Scalar Case 215

9.3.4 Innovation 217

9.3.5 Recursive Bayes Formula – Vector Case 219

9.3.6 Geometric Interpretation of Bayes Estimation 220

9.3.6.1 Geometric Interpretation of the Bayes Batch Formula 220

9.3.6.2 Geometric Interpretation of the Recursive Bayes Formula 222

9.4 One-step-ahead Kalman Predictor 223

9.4.1 The Innovation in the State Prediction Problem 224

9.4.2 The State Prediction Error 224

9.4.3 Optimal One-Step-Ahead Prediction of the Output 225

9.4.4 Optimal One-Step-Ahead Prediction of the State 226

9.4.5 Riccati Equation 228

9.4.6 Initialization 231

9.4.7 One-step-ahead Optimal Predictor Summary 232

9.4.8 Generalizations 236

9.4.8.1 System 236

9.4.8.2 Predictor 236

9.5 Multistep Optimal Predictor 237

9.6 Optimal Filter 239

9.7 Steady-State Predictor 240

9.7.1 Gain Convergence 241

9.7.2 Convergence of the Riccati Equation Solution 244

9.7.2.1 Convergence Under Stability 244

9.7.2.2 ConvergenceWithout Stability 246

9.7.2.3 Observability 250

9.7.2.4 Reachability 251

9.7.2.5 General Convergence Result 256

9.8 Innovation Representation 265

9.9 Innovation Representation Versus Canonical Representation 266

9.10 K-Theory Versus K–W Theory 267

9.11 Extended Kalman Filter – EKF 271

9.12 The Robust Approach to Filtering 273

9.12.1 Norm of a Dynamic System 274

9.12.2 Robust Filtering 276

10 Parameter Identification in a Given Model 281

10.1 Introduction 281

10.2 Kalman Filter-Based Approaches 281

10.3 Two-Stage Method 284

10.3.1 First Stage – Data Generation and Compression 285

10.3.2 Second Stage – Compressed Data Fitting 287

11 Case Studies 291

11.1 Introduction 291

11.2 Kobe Earthquake Data Analysis 291

11.2.1 Modeling the Normal Seismic Activity Data 294

11.2.2 Model Validation 296

11.2.3 Analysis of the Transition Phase via Detection Techniques 299

11.2.4 Conclusions 300

11.3 Estimation of a Sinusoid in Noise 300

11.3.1 Frequency Estimation by Notch Filter Design 301

11.3.2 Frequency Estimation with EKF 305

Appendix A Linear Dynamical Systems 309

A.1 State Space and Input–Output Models 309

A.1.1 Characteristic Polynomial and Eigenvalues 309

A.1.2 Operator Representation 310

A.1.3 Transfer Function 310

A.1.4 Zeros, Poles, and Eigenvalues 310

A.1.5 Relative Degree 311

A.1.6 Equilibrium Point and System Gain 311

A.2 Lagrange Formula 312

A.3 Stability 312

A.4 Impulse Response 313

A.4.1 Impulse Response from a State Space Model 314

A.4.2 Impulse Response from an Input–Output Model 314

A.4.3 Quadratic Summability of the Impulse Response 315

A.5 Frequency Response 315

A.6 Multiplicity of State Space Models 316

A.6.1 Change of Basis 316

A.6.2 Redundancy in the System Order 317

A.7 Reachability and Observability 318

A.7.1 Reachability 318

A.7.2 Observability 320

A.7.3 PBH Test of Reachability and Observability 321

A.8 System Decomposition 323

A.8.1 Reachability and Observability Decompositions 323

A.8.2 Canonical Decomposition 324

A.9 Stabilizability and Detectability 328

Appendix B Matrices 331

B.1 Basics 331

B.2 Eigenvalues 335

B.3 Determinant and Inverse 337

B.4 Rank 340

B.5 Annihilating Polynomial 342

B.6 Algebraic and Geometric Multiplicity 345

B.7 Range and Null Space 345

B.8 Quadratic Forms 346

B.9 Derivative of a Scalar Function with Respect to a Vector 349

B.10 Matrix Diagonalization via Similarity 350

B.11 Matrix Diagonalization via Singular Value Decomposition 351

B.12 Matrix Norm and Condition Number 353

Appendix C Problems and Solutions 357

Bibliography 391

Index 397