Nonparametric system identification

Presenting a thorough overview of the theoretical foundations of non-parametric system identification for nonlinear block-oriented systems, this book shows that non-parametric regression can be successfully applied to system identification, and it highlights the achievements in doing so. With emphasis on Hammerstein, Wiener systems, and their multidimensional extensions, the authors show how to identify nonlinear subsystems and their characteristics when limited information exists. Algorithms using trigonometric, Legendre, Laguerre, and Hermite series are investigated, and the kernel algorithm, its semirecursive versions, and fully recursive modifications are covered. The theories of modern non-parametric regression, approximation, and orthogonal expansions, along with new approaches to system identification (including semiparametric identification), are provided. Detailed information about all tools used is provided in the appendices. This book is for researchers and practitioners in systems theory, signal processing, and communications and will appeal to researchers in fields like mechanics, economics, and biology, where experimental data are used to obtain models of systems.

Preface	p. ix
1 Introduction	p. 1
2 Discrete-time Hammerstein systems	p. 3
2.1 The system	p. 3
2.2 Nonlinear subsystem	p. 4
2.3 Dynamic subsystem identification	p. 8
2.4 Bibliographic notes	p. 9
3 Kernel algorithms	p. 11
3.1 Motivation	p. 11
3.2 Consistency	p. 13
3.3 Applicable kernels	p. 14
3.4 Convergence rate	p. 16
3.5 The mean-squared error	p. 21
3.6 Simulation example	p. 21
3.7 Lemmas and proofs	p. 24
3.8 Bibliographic notes	p. 29
4 Semirecursive kernel algorithms
4.1 Introduction	p. 30
4.2 Consistency and convergence rate	p. 31
4.3 Simulation example	p. 34
4.4 Proofs and lemmas	p. 35
4.5 Bibliographic notes	p. 43
5 Recursive kernel algorithms	p. 44
5.1 Introduction	p. 44
5.2 Relation to stochastic approximation	p. 44
5.3 Consistency and convergence rate	p. 46
5.4 Simulation example	p. 49
5.5 Auxiliary results, lemmas, and proofs	p. 51
5.6 Bibliographic notes	p. 58
6 Orthogonal series algorithms	p. 59
6.1 Introduction	p. 59
6.2 Fourier series estimate	p. 61
6.3 Legendre series estimate	p. 64
6.4 Laguerre series estimate	p. 66
6.5 Hermite series estimate	p. 68
6.6 Wavelet estimate	p. 69
6.7 Local and global errors	p. 70
6.8 Simulation example	p. 71
6.9 Lemmas and proofs	p. 72
6.10 Bibliographic notes	p. 78
7 Algorithms with ordered observations	p. 80
7.1 Introduction	p. 80
7.2 Kernel estimates	p. 81
7.3 Orthogonal series estimates	p. 85
7.4 Lemmas and proofs	p. 89
7.5 Bibliographic notes	p. 99
8 Continuous-time Hammerstein systems	p. 101
8.1 Identification problem	p. 101
8.2 Kernel algorithm	p. 103
8.3 Orthogonal series algorithms	p. 106
8.4 Lemmas and proofs	p. 108
8.5 Bibliographic notes	p. 112
9 Discrete-time Wiener systems	p. 113
9.1 The system	p. 113
9.2 Nonlinear subsystem	p. 114
9.3 Dynamic subsystem identification	p. 119
9.4 Lemmas	p. 121
9.5 Bibliographic notes	p. 122
10 Kernel and orthogonal series algorithms	p. 123
10.1 Kernel algorithms	p. 123
10.2 Orthogonal series algorithms	p. 126
10.3 Simulation example	p. 129
10.4 Lemmas and proofs	p. 130
10.5 Bibliographic notes	p. 142
11 Continuous-time Wiener system	p. 143
11.1 Identification problem	p. 143
11.2 Nonlinear subsystem	p. 144
11.3 Dynamic subsystem	p. 146
11.4 Lemmas	p. 146
11.5 Bibliographic notes	p. 148
12 Other block-oriented nonlinear systems	p. 149
12.1 Series-parallel, block-oriented systems	p. 149
12.2 Block-oriented systems with nonlinear dynamics	p. 173
12.3 Concluding remarks	p. 218
12.4 Bibliographical notes	p. 220
13 Multivariate nonlinear block-oriented systems	p. 222
13.1 Multivariate nonparametric regression	p. 222
13.2 Additive modeling and regression analysis	p. 228
13.3 Multivariate systems	p. 242
13.4 Concluding remarks	p. 248
13.5 Bibliographic notes	p. 248
14 Semiparametric identification	p. 250
14.1 Introduction	p. 250
14.2 Semiparametric models	p. 252
14.3 Statistical inference for semiparametric models	p. 255
14.4 Statistical inference for semiparametric Wiener models	p. 264
14.5 Statistical inference for semiparametric Hammerstein models	p. 286
14.6 Statistical inference for semiparametric parallel models	p. 287
14.7 Direct estimators for semiparametric systems	p. 290
14.8 Concluding remarks	p. 309
14.9 Auxiliary results, lemmas, and proofs	p. 310
14.10 Bibliographical notes	p. 316
A Convolution and kernel functions	p. 319
A.1 Introduction	p. 319
A.2 Convergence	p. 320
A.3 Applications to probability	p. 328
A.4 Lemmas	p. 329
B Orthogonal functions	p. 331
B.1 Introduction	p. 331
B.2 Fourier series	p. 333
B.3 Legendre series	p. 340
B.4 Laguerre series	p. 345
B.5 Hermite series	p. 351
B.6 Wavelets	p. 355
C Probability and statistics	p. 359
C.1 White noise	p. 359
C.2 Convergence of random variables	p. 361
C.3 Stochastic approximation	p. 364
C.4 Order statistics	p. 365
References	p. 371
Index	p. 387

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