An in-depth introduction to subspace methods for system identification in discrete-time linear systems thoroughly augmented with advanced and novel results, this text is structured into three parts. Part I deals with the mathematical preliminaries: numerical linear algebra; system theory; stochastic processes; and Kalman filtering. Part II explains realization theory as applied to subspace identification. Stochastic realization results based on spectral factorization and Riccati equations, and on canonical correlation analysis for stationary processes are included. Part III demonstrates the closed-loop application of subspace identification methods. Subspace Methods for System Identification is an excellent reference for researchers and a useful text for tutors and graduate students involved in control and signal processing courses. It can be used for self-study and will be of interest to applied scientists or engineers wishing to use advanced methods in modeling and identification of complex systems.
Author(s): Tohru Katayama
Edition: 1st Edition.
Year: 2005
Language: English
Pages: 400
1852339810......Page 1
Contents......Page 10
1.1 System Identification......Page 15
1.2 Classical Identification Methods......Page 18
1.3 Prediction Error Method for State Space Models......Page 20
1.4 Subspace Methods of System Identification......Page 22
1.5 Historical Remarks......Page 25
1.6 Outline of the Book......Page 27
1.7 Notes and References......Page 28
Part I Preliminaries......Page 29
2.1 Vectors and Matrices......Page 30
2.2 Subspaces and Linear Independence......Page 32
2.3 Norms of Vectors and Matrices......Page 34
2.4 QR Decomposition......Page 36
2.5 Projections and Orthogonal Projections......Page 40
2.6 Singular Value Decomposition......Page 43
2.7 Least-Squares Method......Page 46
2.8 Rank of Hankel Matrices......Page 49
2.9 Notes and References......Page 51
2.10 Problems......Page 52
3.1 z-Transform......Page 54
3.2 Discrete-Time LTI Systems......Page 57
3.3 Norms of Signals and Systems......Page 60
3.4 State Space Systems......Page 61
3.5 Lyapunov Stability......Page 63
3.6 Reachability and Observability......Page 64
3.7 Canonical Decomposition of Linear Systems......Page 68
3.8 Balanced Realization and Model Reduction......Page 71
3.9 Realization Theory......Page 78
3.10 Notes and References......Page 83
3.11 Problems......Page 84
4.1 Stochastic Processes......Page 86
4.1.1 Markov Processes......Page 87
4.1.2 Means and Covariance Matrices......Page 88
4.2 Stationary Stochastic Processes......Page 90
4.3 Ergodic Processes......Page 92
4.4 Spectral Analysis......Page 94
4.5 Hilbert Space and Prediction Theory......Page 100
4.6 Stochastic Linear Systems......Page 108
4.7 Stochastic Linear Time-Invariant Systems......Page 111
4.8 Backward Markov Models......Page 114
4.9 Notes and References......Page 117
4.10 Problems......Page 118
5.1 Multivariate Gaussian Distribution......Page 120
5.2 Optimal Estimation by Orthogonal Projection......Page 126
5.3 Prediction and Filtering Algorithms......Page 129
5.4 Kalman Filter with Inputs......Page 136
5.5 Covariance Equation of Predicted Estimate......Page 140
5.6 Stationary Kalman Filter......Page 141
5.7 Stationary Backward Kalman Filter......Page 144
5.8 Numerical Solution of ARE......Page 147
5.9 Notes and References......Page 149
5.10 Problems......Page 150
Part II Realization Theory......Page 152
6.1 Realization Problems......Page 153
6.2 Ho-Kalman’s Method......Page 154
6.3 Data Matrices......Page 161
6.4 LQ Decomposition......Page 167
6.5 MOESP Method......Page 169
6.6 N4SID Method......Page 173
6.7 SVD and Additive Noises......Page 178
6.8 Notes and References......Page 181
6.9 Problems......Page 182
7.1 Preliminaries......Page 183
7.2 Stochastic Realization Problem......Page 186
7.3 Solution of Stochastic Realization Problem......Page 188
7.3.1 Linear Matrix Inequality......Page 189
7.3.2 Simple Examples......Page 192
7.4.1 Positive Real Lemma......Page 195
7.4.2 Computation of Extremal Points......Page 201
7.5 Algebraic Riccati-like Equations......Page 204
7.6 Strictly Positive Real Conditions......Page 206
7.7 Stochastic Realization Algorithm......Page 210
7.8 Notes and References......Page 211
7.9 Problems......Page 212
7.10 Appendix: Proof of Lemma 7.4......Page 213
8.1 Canonical Correlation Analysis......Page 215
8.2 Stochastic Realization Problem......Page 219
8.3.1 Predictor Spaces......Page 221
8.3.2 Markovian Representations......Page 224
8.4 Canonical Correlations Between Future and Past......Page 228
8.5.1 Forward and Backward State Vectors......Page 229
8.5.2 Innovation Representations......Page 231
8.6 Reduced Stochastic Realization......Page 235
8.7 Stochastic Realization Algorithms......Page 239
8.8 Numerical Results......Page 242
8.9 Notes and References......Page 244
8.10 Problems......Page 245
8.11 Appendix: Proof of Lemma 8.5......Page 246
Part III Subspace Identification......Page 249
9.1 Projections......Page 250
9.2 Stochastic Realization with Exogenous Inputs......Page 252
9.3 Feedback-Free Processes......Page 254
9.4.1 Orthogonal Decomposition......Page 256
9.4.2 PE Condition......Page 257
9.5.1 Realization of Stochastic Component......Page 259
9.5.2 Realization of Deterministic Component......Page 260
9.5.3 The Joint Model......Page 264
9.6 Realization Based on Finite Data......Page 265
9.7.1 Subspace Identification of Deterministic Subsystem......Page 267
9.7.2 Subspace Identification of Stochastic Subsystem......Page 270
9.8 Numerical Example......Page 272
9.10.1 Proof of Theorem 9.1......Page 276
9.10.2 Proof of Lemma 9.7......Page 279
10.1 Stochastic Realization with Exogenous Inputs......Page 282
10.2 Optimal Predictor......Page 286
10.3 Conditional Canonical Correlation Analysis......Page 289
10.4 Innovation Representation......Page 293
10.5 Stochastic Realization Based on Finite Data......Page 297
10.6 CCA Method......Page 299
10.7 Numerical Examples......Page 303
10.8 Notes and References......Page 307
11.1 Overview of Closed-loop Identification......Page 310
11.2.1 Feedback System......Page 312
11.2.2 Identification by Joint Input-Output Approach......Page 314
11.3.1 Realization of Joint Input-Output Process......Page 315
11.3.2 Subspace Identification Method......Page 318
11.4.1 Orthogonal Decomposition of Joint Input-Output Process......Page 320
11.4.2 Realization of Closed-loop System......Page 322
11.4.3 Subspace Identification Method......Page 323
11.5 Model Reduction......Page 326
11.6 Numerical Results......Page 328
11.6.1 Example 1......Page 329
11.6.2 Example 2......Page 332
11.7 Notes and References......Page 334
11.8.1 Identification of Deterministic Parts......Page 335
11.8.2 Identification of Noise Models......Page 336
Appendix......Page 338
A.1 Linear Regressions......Page 339
A.2 LQ Decomposition......Page 344
B Input Signals for System Identification......Page 347
C Overlapping Parametrization......Page 353
D.1 Deterministic Realization Algorithm......Page 359
D.2 MOESP Algorithm......Page 360
D.3 Stochastic Realization Algorithms......Page 361
D.4 Subspace Identification Algorithms......Page 363
E Solutions to Problems......Page 366
Glossary......Page 386
References......Page 388
D......Page 397
M......Page 398
R......Page 399
Z......Page 400
