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EE EE Con
- ntrol Systems
tems
Identification of Nonlinear LFR Systems starting from the Best Linear Approximation
- M. Schoukens and R. TΓ³th
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Identification of Nonlinear LFR Systems starting from the Best - - PowerPoint PPT Presentation
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. Tth EE EE Con ontrol Systems tems Nonlinear System Class 2 Outline Nonlinear System Class Initialization & Estimation Examples
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Nonlinear System Class Initialization & Estimation Examples Conclusions
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Outline
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Nonlinear System Class Initialization & Estimation Examples Conclusions
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Outline
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Nonlinear System Class
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Nonlinear System Class
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Nonlinear LFR vs Nonlinear SS
Structured NL State-Space
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Nonlinear LFR vs Nonlinear SS
Full NL State-Space πΆw = π½ππ¦ π·z = π½ππ¦ 0ππ£ πΈzu = 0ππ¦ π½ππ£ πΈyw = π½ππ§
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Uniqueness of the Representation
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Uniqueness of the Representation
All the problems of linear state-space representation
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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
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Nonlinear System Class Initialization & Estimation Examples Conclusions
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Outline
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Initialization & Estimation
Step 1: Estimate the Best Linear Approximation
Initial estimate of:
Frequency Domain Nonparametric BLA Rational Transfer Function State-Space Realization
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Initialization & Estimation
Step 1: Estimate the Best Linear Approximation
For a good initial estimate, all the states should be βvisibleβ for the best linear approximation of the system
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Initialization & Estimation
Step 2: Nonlinear optimization of all the parameters together Initializing Nonlinearity, w and z Matrices Nonlinearity can be replaced in a 3rd step
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Initialization & Estimation
Step 2: Nonlinear optimization of all the parameters together
Nonlinear Optimization Levenberg-Marquardt Optimization
simulation error
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Nonlinear System Class Initialization & Estimation Examples Conclusions
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Outline
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Silverbox Benchmark
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Silverbox Benchmark
Validation Estimation
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Silverbox Benchmark
nx = 2 3rd degree polynomial nonlinearity
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Silverbox Benchmark
nx = 2 3rd 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
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Silverbox Benchmark
Validation Estimation
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Silverbox Benchmark
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Silverbox Benchmark
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Silverbox Benchmark
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Wiener-Hammerstein Benchmark
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Wiener-Hammerstein Benchmark
Validation Estimation
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Wiener-Hammerstein Benchmark
nx = 6 5th degree polynomial nonlinearity Neural network 20 neurons β 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
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Wiener-Hammerstein Benchmark
Validation Estimation
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Wiener-Hammerstein Benchmark
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Wiener-Hammerstein Benchmark
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Wiener-Hammerstein Benchmark
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Wiener-Hammerstein Benchmark
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Nonlinear System Class Initialization & Estimation Examples Conclusions
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Outline
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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
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EE EE Con
- ntrol Systems
tems
Identification of Nonlinear LFR Systems starting from the Best Linear Approximation
- M. Schoukens and R. TΓ³th