Identification of Wiener-Hammerstein systems with process noise - - PowerPoint PPT Presentation

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Mar 10, 2023 369 likes •695 views

Identification of Wiener-Hammerstein systems with process noise using an Errors-in-Variables framework Maarten Schoukens, Fritjof Griesing Scheiwe Benchmarks Overview Best Linear Approximation Wiener-Hammerstein Output-Error Influence of

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Identification of Wiener-Hammerstein systems with process noise using an Errors-in-Variables framework

Maarten Schoukens, Fritjof Griesing Scheiwe

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Benchmarks

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Overview

Best Linear Approximation Wiener-Hammerstein – Output-Error Influence of the process noise? Wiener-Hammerstein – EIV Results

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Overview

Best Linear Approximation Wiener-Hammerstein – Output-Error Influence of the process noise? Wiener-Hammerstein – EIV Results

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Best Linear Approximation

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Bussgang’s Theorem

Stationary Gaussian input  Static nonlinearity ≈ static gain

𝑔 𝑣 = 𝛿𝑣

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Structure detection

Wiener-Hammerstein  Only gain factor

) ( ) ( ) ( q S q H q Gbla  

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Overview

Best Linear Approximation Wiener-Hammerstein – Output-Error Influence of the process noise? Wiener-Hammerstein – EIV Results

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Wiener-Hammerstein: OE

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Identifiability

Gain exchange

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Identifiability

Gain exchange Delay exchange

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Best Linear Approximation

) ( ) ( ) ( q S q H q Gbla  

 poles, zeros BLA = poles, zeros system

Gaussian

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Partition the Dynamics

BLA

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Nonlinear optimization

Initial parameter values  Optimization of all parameters together  Levenberg-Marquardt algorithm

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Overview

Best Linear Approximation Wiener-Hammerstein – Output-Error Influence of the process noise? Wiener-Hammerstein – EIV Results

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Influence of the Process Noise

ex(t): Gaussian x(t): Gaussian Bussgang’s Theorem

) ( ) ( ) ( s R s S s Gbla  

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Influence of the Process Noise

ex(t): Gaussian x(t): Gaussian Bussgang’s Theorem

) ( ) ( ) ( s R s S s Gbla  

x(t) depends on ex(t) γ depends on ex(t)

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Example: 3rd Degree NL

𝑧 = 𝑦0 + 𝑓𝑦 3 = 𝑦03 + 3𝑓𝑦𝑦02 + 3𝑓𝑦2𝑦0 + 𝑓𝑦3 𝛿 = 𝐹 𝑧𝑦0 𝐹 𝑦0𝑦0 = 𝐹 𝑦04 + 3𝑓𝑦𝑦03 + 3𝑓𝑦2𝑦02 + 𝑓𝑦3𝑦0 𝐹 𝑦02 = 𝐹 𝑦04 + 3𝑓𝑦2𝑦02 𝐹 𝑦02 = 3𝜏𝑦4 + 3𝜏𝑦2𝜏𝑓2 𝜏𝑦2 = 3𝜏𝑦2 + 3𝜏𝑓2

Bias due to odd nonlinear terms

Assumptions: Gaussian Zero-mean Independent

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Overview

Best Linear Approximation Wiener-Hammerstein – Output-Error Influence of the process noise? Wiener-Hammerstein – EIV Results

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Wiener-Hammerstein: EIV

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Wiener-Hammerstein: EIV

EIV Framework?

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Wiener-Hammerstein: EIV

EIV Framework?

Output depends on input noise  Bias!

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Wiener-Hammerstein: EIV

EIV Framework?

Let us try it anyway:

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Wiener-Hammerstein: EIV

Let us try it anyway: Direct optimization of input on selected freq. Penalty term introduces prior knowledge

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Overview

Best Linear Approximation Wiener-Hammerstein – Output-Error Influence of the process noise? Wiener-Hammerstein – EIV Results

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Results

Estimation Data Random Phase Multisine Input: frequencies: 0-15 kHz RMS: 0.7581 4096 Samples 1 Period 10 Realizations fs: 78.125 kHz

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Results

BLA: order 6/6 Wiener-Hammerstein: Neural Network 3 tansig activation functions

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Results

Output Linear Error WH EIV

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Results

Output

Linear Error WH EIV

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Results

Simulation – Validation/Test Results Multisine

LTI WH OE WH EIV Realization 1 0.055875 0.031387 0.022458 Realization 2 0.055875 0.031387 0.023080 Realization 3 0.055875 0.031387 0.040724 Realization 1-3 0.055875 0.031113 0.025004

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Results

Simulation – Validation/Test Results Sinesweep

LTI WH OE WH EIV Realization 1 0.019492 0.01485 0.039964 Realization 2 0.019492 0.01485 0.02139 Realization 3 0.019492 0.01485 0.022443 Realization 1-3 0.019492 0.011967 0.01913

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Conclusions

Process noise introduces a bias through odd NL terms Identification with process noise is not just an EIV problem EIV methods can result in better estimates