Identification of Nonlinear LFR Systems starting from the Best - PowerPoint PPT Presentation

Identification of Nonlinear LFR Systems starting from the Best Linear Approximation M. Schoukens and R. Tóth EE EE Con ontrol Systems tems

Nonlinear System Class 2

Outline Nonlinear System Class Initialization & Estimation Examples Conclusions 3

Outline Nonlinear System Class Initialization & Estimation Examples Conclusions 4

Nonlinear System Class 5

Nonlinear System Class 6

Nonlinear LFR vs Nonlinear SS Structured NL State-Space 7

Nonlinear LFR vs Nonlinear SS 𝐶 w = 𝐽𝑜𝑦 𝐷 z = 𝐽 𝑜𝑦 𝐸 zu = 0 𝑜𝑦 0 𝑜𝑣 𝐸 yw = 𝐽𝑜𝑧 𝐽 𝑜𝑣 Full NL State-Space 8

Uniqueness of the Representation 9

Uniqueness of the Representation All the problems of linear state-space representation 10

Uniqueness of the Representation All the problems of linear state-space representation Additional exchange of a linear gain between the nonlinearity and the linear dynamics 11

Outline Nonlinear System Class Initialization & Estimation Examples Conclusions 12

Initialization & Estimation Step 1: Estimate the Best Linear Approximation Frequency Domain Nonparametric BLA Initial estimate of: Rational Transfer Function State-Space Realization 13

Initialization & Estimation Step 1: Estimate the Best Linear Approximation For a good initial estimate, all the states should be ‘visible’ for the best linear approximation of the system 14

Initialization & Estimation Step 2: Nonlinear optimization of all the parameters together Initializing Nonlinearity, w and z Matrices Nonlinearity can be replaced in a 3 rd step 15

Initialization & Estimation Step 2: Nonlinear optimization of all the parameters together Nonlinear Optimization simulation error Levenberg-Marquardt Optimization 16

Outline Nonlinear System Class Initialization & Estimation Examples Conclusions 17

Silverbox Benchmark 18

Silverbox Benchmark Validation Estimation 19

Silverbox Benchmark n x = 2 3 rd degree polynomial nonlinearity 20

Silverbox Benchmark n x = 2 3 rd degree polynomial nonlinearity rms errors on estimation data linear model error: 6.62 mV NL-LFR error: 0.25 mV rms errors on validation data linear model error: 14.5 mV NL-LFR error: 0.38 mV 21

Silverbox Benchmark Validation Estimation 22

Silverbox Benchmark 23

Silverbox Benchmark 24

Silverbox Benchmark 25

Wiener-Hammerstein Benchmark 26

Wiener-Hammerstein Benchmark Estimation Validation 27

Wiener-Hammerstein Benchmark n x = 6 5 th degree polynomial nonlinearity Neural network 20 neurons – 1 hidden layer - tansig rms errors on estimation data linear model error: 55.8 mV NL-LFR error: 0.29 mV rms errors on validation data linear model error: 56.1 mV NL-LFR error: 0.30 mV 28

Wiener-Hammerstein Benchmark Estimation Validation 29

Wiener-Hammerstein Benchmark 30

Wiener-Hammerstein Benchmark 31

Wiener-Hammerstein Benchmark 32

Wiener-Hammerstein Benchmark 33

Outline Nonlinear System Class Initialization & Estimation Examples Conclusions 34

Conclusions Structured model directly from the data Linear initial model followed by NL optimization Good results on simple benchmark examples Future work: MIMO NL, MIMO LTI 35

Identification of Nonlinear LFR Systems starting from the Best Linear Approximation M. Schoukens and R. Tóth EE EE Con ontrol Systems tems