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E-grāmata: Combined Parametric-Nonparametric Identification of Block-Oriented Systems

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Combined Parametric-Nonparametric Identification of Block-Oriented SystemsPalielināt
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This book considers a problem of block-oriented nonlinear dynamic system identification in the presence of random disturbances. This class of systems includes various interconnections of linear dynamic blocks and static nonlinear elements, e.g., Hammerstein system, Wiener system, Wiener-Hammerstein ("sandwich") system and additive NARMAX systems with feedback. Interconnecting signals are not accessible for measurement. The combined parametric-nonparametric algorithms, proposed in the book, can be selected dependently on the prior knowledge of the system and signals. Most of them are based on the decomposition of the complex system identification task into simpler local sub-problems by using non-parametric (kernel or orthogonal) regression estimation. In the parametric stage, the generalized least squares or the instrumental variables technique is commonly applied to cope with correlated excitations. Limit properties of the algorithms have been shown analytically and illustrated in simple experiments.
Preface vii
Abstract ix
1 Introduction
1(22)
1.1 Parametric Identification of Single-Element Systems
1(9)
1.1.1 Least Squares Approach
1(8)
1.1.2 Instrumental Variables Method
9(1)
1.2 Nonparametric Methods
10(3)
1.2.1 Cross-Correlation Analysis
10(1)
1.2.2 Kernel Regression Function Estimation
11(1)
1.2.3 Orthogonal Expansion
12(1)
1.3 Example
13(1)
1.4 Block-Oriented Systems
14(9)
1.4.1 Hammerstein System
14(2)
1.4.2 Wiener System
16(2)
1.4.3 Sandwich (Wiener-Hammerstein) System
18(1)
1.4.4 NARMAX System
19(1)
1.4.5 Interconnected MIMO System
20(1)
1.4.6 Applications in Science and Technology
21(2)
2 Hammerstein System
23(64)
2.1 Finite Memory Hammerstein System
23(10)
2.1.1 Introduction
23(1)
2.1.2 Statement of the Problem
24(1)
2.1.3 Background of the Approach
25(2)
2.1.4 Two-Stage Parametric-Nonparametric Estimation of Nonlinearity Parameters
27(2)
2.1.5 Two-Stage Parametric-Nonparametric Identification of Linear Dynamics
29(1)
2.1.6 Example
30(1)
2.1.7 Simulation Study
31(2)
2.1.8 Conclusions
33(1)
2.2 Infinite Memory Hammerstein System
33(10)
2.2.1 Introduction
33(1)
2.2.2 Statement of the Problem
34(2)
2.2.3 Identification of ARMA Linear Dynamics by Nonparametric Instrumental Variables
36(6)
2.2.4 Conclusions
42(1)
2.3 Hammerstein System with Hard Nonlinearity
43(13)
2.3.1 Parametric Prior Knowledge of the Nonlinear Characteristic
43(3)
2.3.2 Estimation of the Nonlinearity Parameters
46(5)
2.3.3 Simulation Examples
51(5)
2.4 Hammerstein System Excited by Correlated Process
56(14)
2.4.1 Introduction
56(2)
2.4.2 Assumptions
58(1)
2.4.3 Examples
59(2)
2.4.4 Statement of the Problem
61(1)
2.4.5 Semi-parametric Least Squares in Case of Correlated Input
62(1)
2.4.6 Instrumental Variables Method
63(1)
2.4.7 Generation of Instruments
64(2)
2.4.8 Various Kinds of Prior Knowledge
66(2)
2.4.9 Computer Experiment
68(2)
2.4.10 Summary
70(1)
2.5 Cascade Systems
70(6)
2.5.1 Introduction
70(1)
2.5.2 Statement of the Problem
70(1)
2.5.3 Least Squares
71(2)
2.5.4 Nonparametric Generation of Instruments
73(2)
2.5.5 Computer Experiment
75(1)
2.5.6 Summary
76(1)
2.6 Identification under Small Number of Data
76(6)
2.6.1 Introduction
77(1)
2.6.2 Statement of the Problem
77(1)
2.6.3 The Proposed Algorithm
78(3)
2.6.4 Numerical Example
81(1)
2.6.5 Conclusions
81(1)
2.7 Semiparametric Approach
82(3)
2.7.1 Nonlinearity Recovering
82(1)
2.7.2 Identification of the Linear Dynamic Subsystem
83(2)
2.8 Final Remarks
85(2)
3 Wiener System
87(16)
3.1 Introduction to Wiener System
87(1)
3.2 Nonparametric Identification Tools
88(8)
3.2.1 Inverse Regression Approach
88(2)
3.2.2 Cross-Correlation Analysis
90(1)
3.2.3 A Censored Sample Mean Approach
90(6)
3.3 Combined Parametric-Nonparametric Approach
96(6)
3.3.1 Kernel Method with the Correlation-Based Internal Signal Estimation (FIR)
96(1)
3.3.2 Identification of IIR Wiener Systems with Non-Gaussian Input
97(3)
3.3.3 Averaged Derivative Method
100(1)
3.3.4 Mixed Approach
101(1)
3.4 Final Remarks
102(1)
4 Wiener-Hammerstein (Sandwich) System
103(10)
4.1 Introduction
103(1)
4.2 Preliminaries
104(2)
4.3 Assumptions
106(1)
4.4 The Algorithms
107(3)
4.5 Numerical Example
110(1)
4.6 Final Remarks
111(2)
5 Additive NARMAX System
113(24)
5.1 Statement of the Problem
113(3)
5.1.1 The System
113(3)
5.1.2 Comments to Assumptions
116(1)
5.1.3 Organization of the
Chapter
116(1)
5.2 Least Squares and SVD Approach
116(3)
5.3 Instrumental Variables Approach
119(1)
5.4 Limit Properties
120(1)
5.5 Optimal Instrumental Variables
120(8)
5.6 Nonparametric Generation of Instrumental Variables
128(3)
5.7 The 3-Stage Identification
131(1)
5.8 Example
132(4)
5.8.1 Simulation
132(1)
5.8.2 Identification
133(3)
5.9 Final Remarks
136(1)
6 Large-Scale Interconnected Systems
137(12)
6.1 Introduction
137(1)
6.2 Statement of the Problem
138(2)
6.3 Least Squares Approach
140(1)
6.4 Instrumental Variables Approach
141(3)
6.5 Nonlinear Dynamic Components
144(2)
6.6 Simulation Example
146(1)
6.7 Final Remarks
146(3)
7 Structure Detection and Model Order Selection
149(22)
7.1 Introduction
149(2)
7.1.1 Types of Knowledge. Classification of Approaches
149(1)
7.1.2 Contribution
150(1)
7.2 Statement of the Problem
151(2)
7.2.1 The System
151(1)
7.2.2 Assumptions
151(1)
7.2.3 Preliminaries
152(1)
7.3 Parametric Approximation of the Regression Function
153(4)
7.3.1 The Best Model
153(1)
7.3.2 Approximation of the Regression Function
154(2)
7.3.3 The Nonlinear Least Squares Method
156(1)
7.4 Model Quality Evaluation
157(1)
7.5 The Algorithm of the Best Model Selection
158(2)
7.6 Limit Properties
160(2)
7.7 Computational Complexity
162(1)
7.8 Examples
162(6)
7.8.1 Example I -- Voltage-Intensity Diode Characteristic
162(2)
7.8.2 Example II -- Model Order Selection
164(1)
7.8.3 Example III -- Fast Polynomial vs. Piecewise-Linear Model Competition
164(4)
7.9 Final Remarks
168(3)
8 Time-Varying Systems
171(10)
8.1 Introduction
171(1)
8.2 Kernel Estimate for Time-Varying Regression
172(3)
8.3 Weighted Least Squares for NARMAX Systems
175(2)
8.4 Nonparametric Identification of Periodically Varying Systems
177(1)
8.5 Detection of Structure Changes
178(2)
8.6 Final Remarks
180(1)
9 Simulation Studies
181(8)
9.1 IFAC Wiener-Hammerstein Benchmark
181(1)
9.2 Combined Methods in Action
182(2)
9.3 Results of Experiments
184(2)
9.4 Conclusions
186(3)
10 Summary
189(4)
10.1 Index of Considered Problems
189(2)
10.2 A Critical Look at the Combined Approach
191(2)
A Proofs of Theorems and Lemmas
193(28)
A.1 Recursive Least Squares
193(2)
A.2 Covariance Matrix of the LS Estimate
195(1)
A.3 Hammerstein System as a Special Case of NARMAX System
196(1)
A.4 Proof of Theorem 2.1
197(1)
A.5 Proof of Theorem 2.2
198(1)
A.6 Proof of Theorem 2.3
198(2)
A.7 Proof of Theorem 2.4
200(1)
A.8 Proof of Theorem 2.5
201(1)
A.9 Proof of Theorem 2.7
202(3)
A.10 Proof of Theorem 2.8
205(1)
A.11 Proof of Theorem 2.9
206(1)
A.12 Proof of Theorem 2.10
207(1)
A.13 Proof of Theorem 2.11
207(1)
A.14 Proof of Theorem 2.13
208(2)
A.15 Proof of Proposition 3.1
210(1)
A.16 Proof of Theorem 3.3
210(1)
A.17 Proof of Theorem 4.1
211(3)
A.18 The Necessary Condition for the 3-Stage Algorithm
214(1)
A.19 Proof of Theorem 5.1
215(1)
A.20 Proof of Theorem 5.2
215(2)
A.21 Proof of Theorem 5.4
217(2)
A.22 Proof of Theorem 5.7
219(1)
A.23 Example of ARX(1) Object
220(1)
B Algebra Toolbox
221(6)
B.1 SVD Decomposition
221(1)
B.2 Factorization Theorem
221(1)
B.3 Technical Lemmas
222(1)
B.4 Types of Convergence of Random Sequences
223(1)
B.5 Slutzky Theorem
224(1)
B.6 Chebychev's Inequality
224(1)
B.7 Persistent Excitation
225(1)
B.8 Ergodic Processes
225(1)
B.9 Modified Triangle Inequality
226(1)
References 227(10)
Index 237
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