Filtering and System Identification: A Least Squares Approach

by Michel Verhaegen , Vincent Verdult

• ISBN13: 9780521875127
• ISBN10: 0521875129
• Edition: 1st
• Format: Hardcover
• Copyright: 2007-05-28
• Publisher: Cambridge University Press

Summary

Filtering and system identification are powerful techniques for building models of complex systems. This book discusses the design of reliable numerical methods to retrieve missing information in models derived using these techniques. Emphasis is on the least squares approach as applied to the linear state-space model, and problems of increasing complexity are analyzed and solved within this framework, starting with the Kalman filter and concluding with the estimation of a full model, noise statistics and state estimator directly from the data. Key background topics, including linear matrix algebra and linear system theory, are covered, followed by different estimation and identification methods in the state-space model. With end-of-chapter exercises, MATLAB simulations and numerous illustrations, this book will appeal to graduate students and researchers in electrical, mechanical, and aerospace engineering. It is also useful for practitioners. Additional resources for this title, including solutions for instructors, are available online at www.cambridge.org/9780521875127.

Author Biography

Michel Verhaegen is Professor and co-director of the Delft Center for Systems and Control at the Delft University of Technology in the Netherlands Vincent Verdult is an Assistant Professor in the Delft Centre for Systems and Control at the Delft University of Technology in the Netherlands

Table of Contents

Preface	p. xi
Notation and symbols	p. xiii
List of abbreviations	p. xv
Introduction	p. 1
Linear algebra	p. 8
Introduction	p. 8
Vectors	p. 9
Matrices	p. 13
Square matrices	p. 18
Matrix decompositions	p. 25
Linear least-squares problems	p. 28
Solution if the matrix F has full column rank	p. 32
Solutions if the matrix F does not have full column rank	p. 33
Weighted linear least-squares problems	p. 35
Summary	p. 37
Discrete-time signals and systems	p. 42
Introduction	p. 42
Signals	p. 43
Signal transforms	p. 47
The z-transform	p. 47
The discrete-time Fourier transform	p. 50
Linear systems	p. 55
Linearization	p. 58
System response and stability	p. 59
Controllability and observability	p. 64
Input-output descriptions	p. 69
Interaction between systems	p. 78
Summary	p. 82
Random variables and signals	p. 87
Introduction	p. 87
Description of a random variable	p. 88
Experiments and events	p. 90
The probability model	p. 90
Linear functions of a random variable	p. 95
The expected value of a random variable	p. 95
Gaussian random variables	p. 96
Multiple random variables	p. 97
Random signals	p. 100
Expectations of random signals	p. 100
Important classes of random signals	p. 101
Stationary random signals	p. 102
Ergodicity and time averages of random signals	p. 104
Power spectra	p. 105
Properties of least-squares estimates	p. 108
The linear least-squares problem	p. 109
The weighted linear least-squares problem	p. 112
The stochastic linear least-squares problem	p. 113
A square-root solution to the stochastic linear least-squares problem	p. 115
Maximum-likelihood interpretation of the weighted linear least-squares problem	p. 120
Summary	p. 121
Kalman filtering	p. 126
Introduction	p. 127
The asymptotic observer	p. 128
The Kalman-filter problem	p. 133
The Kalman filter and stochastic least squares	p. 135
The Kalman filter and weighted least squares	p. 141
A weighted least-squares problem formulation	p. 141
The measurement update	p. 142
The time update	p. 146
The combined measurement-time update	p. 150
The innovation form representation	p. 152
Fixed-interval smoothing	p. 159
The Kalman filter for LTI systems	p. 162
The Kalman filter for estimating unknown inputs	p. 166
Summary	p. 171
Estimation of spectra and frequency-response functions	p. 178
Introduction	p. 178
The discrete Fourier transform	p. 180
Spectral leakage	p. 185
The FFT algorithm	p. 188
Estimation of signal spectra	p. 191
Estimation of FRFs and disturbance spectra	p. 195
Periodic input sequences	p. 196
General input sequences	p. 198
Estimating the disturbance spectrum	p. 200
Summary	p. 203
Output-error parametric model estimation	p. 207
Introduction	p. 207
Problems in estimating parameters of an LTI state-space model	p. 209
Parameterizing a MIMO LTI state-space model	p. 213
The output normal form	p. 219
The tridiagonal form	p. 226
The output-error cost function	p. 227
Numerical parameter estimation	p. 231
The Gauss-Newton method	p. 233
Regularization in the Gauss-Newton method	p. 237
The steepest descent method	p. 237
Gradient projection	p. 239
Analyzing the accuracy of the estimates	p. 242
Dealing with colored measurement noise	p. 245
Weighted least squares	p. 247
Prediction-error methods	p. 248
Summary	p. 248
Prediction-error parametric model estimation	p. 254
Introduction	p. 254
Prediction-error methods for estimating state-space models	p. 256
Parameterizing an innovation state-space model	p. 257
The prediction-error cost function	p. 259
Numerical parameter estimation	p. 263
Analyzing the accuracy of the estimates	p. 264
Specific model parameterizations for SISO systems	p. 265
The ARMAX and ARX model structures	p. 266
The Box-Jenkins and output-error model structures	p. 271
Qualitative analysis of the model bias for SISO systems	p. 275
Estimation problems in closed-loop systems	p. 283
Summary	p. 286
Subspace model identification	p. 292
Introduction	p. 292
Subspace model identification for deterministic systems	p. 294
The data equation	p. 294
Identification for autonomous systems	p. 297
Identification using impulse input sequences	p. 299
Identification using general input sequences	p. 301
Subspace identification with white measurement noise	p. 307
The use of instrumental variables	p. 312
Subspace identification with colored measurement noise	p. 315
Subspace identification with process and measurement noise	p. 321
The PO-MOESP method	p. 326
Subspace identification as a least-squares problem	p. 329
Estimating the Kalman gain K[subscript T]	p. 333
Relations among different subspace identification methods	p. 334
Using subspace identification with closed-loop data	p. 336
Summary	p. 338
The system-identification cycle	p. 345
Introduction	p. 346
Experiment design	p. 349
Choice of sampling frequency	p. 349
Transient-response analysis	p. 352
Experiment duration	p. 355
Persistency of excitation of the input sequence	p. 356
Types of input sequence	p. 366
Data pre-processing	p. 369
Decimation	p. 369
Detrending the data	p. 370
Pre-filtering the data	p. 372
Concatenating data sequences	p. 373
Selection of the model structure	p. 373
Delay estimation	p. 373
Model-structure selection in ARMAX model estimation	p. 376
Model-structure selection in subspace identification	p. 382
Model validation	p. 387
The auto-correlation test	p. 388
The cross-correlation test	p. 388
The cross-validation test	p. 390
Summary	p. 390
References	p. 395
Index	p. 401
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