Fast location of the process noise for nonlinear system - - PowerPoint PPT Presentation

Fast location of the process noise for nonlinear system identification

Erliang Zhang*, Maarten Schoukens**

*School of Mechanical Engineering, Zhengzhou University, China ** ELEC, Vrije Universiteit Brussel, Belgium

April 2017

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Workshop on Nonlinear System Identification Benchmarks - April 24-26 2017, Brussels, Belgium

Content

Problem statement Assumptions Previous work Basic idea & Principle Examples Conclusion

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nonlinear systems in the real world

Process noise, output noise & sensor noise Wrong noise model  bias BUT: Including flexible noise structure: increased computational complexity GOAL: Detect process noise in NLSS, NARMAX, Block-Oriented type of structures

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Assumptions

Measurement and process noise

zero-mean stationary noises independent of input excitation

Nonlinear system

Periodic in, same period out Static nonlinearity approximated by polynomial

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     

f

n i i i i i

f t a x t a x t

  



 

i

=

Previous work

Input design: periodic but nonstationary within one period

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Experiment protocol: M experiments, P periods

Zhang, Erliang; Schoukens, Maarten; Schoukens, Joannes , Structure detection of Wiener- Hammerstein systems with process noise. IEEE Transactions on Instrumentation and Measurement, 66(3): 569 – 576, 2017.

Input signal Indicator

• f

the saturation nonlinearity through which the process noise goes in a Wiener-Hammerstein model

Basic idea & principle

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“Specific” input

narrow-band periodic signal in the pass band of the nonlinear system

Connection to the previous work

The amplitude of the sine signal can be seen as the expected value of the random input which is periodic but nonstationary within one period

Aim

Detect the model structure of nonlinear systems with process noise using only one experiment

Basic idea & principle

System response

Case 1: Case 2: ൞ 𝑦𝑙+1 = 𝐵𝑦𝑙 + 𝐶𝑣𝑙 + 𝑔 𝑦𝑙 𝑧𝑙 = 𝐷𝑦𝑙 + 𝐸𝑣𝑙 + 𝑜𝑙 ൞ 𝑦𝑙+1 = 𝐵𝑦𝑙 + 𝐶𝑣𝑙 + 𝑔 𝑦𝑙 + 𝑓𝑙 𝑧𝑙 = 𝐷𝑦𝑙 + 𝐸𝑣𝑙 + 𝑜𝑙

System response when the process noise does pass through the nonlinearity part (Case 1)

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Basic idea & principle

δk is function of the state variable. 𝑑𝑗𝑘 = 𝑗! 𝑗 − 𝑘 ! 𝑘! 𝑦𝑙+1 = ҧ 𝑦𝑙+1 + 𝜀𝑙+1 𝑧𝑙 = 𝐷 ҧ 𝑦𝑙 + 𝐸𝑣𝑙 + 𝐷𝜀𝑙 + 𝑜𝑙 𝑦𝑙+1 = 𝐵 ҧ 𝑦𝑙 + 𝐶𝑣𝑙 + ෍

𝑗=0 𝑜𝑔

𝑏𝑗 ҧ 𝑦𝑙

𝑗

𝜀𝑙+1 = 𝐵𝜀𝑙 + 𝑓𝑙 + ෍

𝑗=1 𝑜𝑔

𝑏𝑗 ෍

𝑘=0 𝑗−1

𝑑𝑗𝑘 ҧ 𝑦𝑙

𝑘𝜀𝑙 𝑗−𝑘

System response when the process noise does not pass through the nonlinearity part (Case 2)

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Basic idea & principle

Remark that the error caused by the noise is independent of the state variable. ൞ 𝑦𝑙+1 = 𝐵𝑦𝑙 + 𝐶𝑣𝑙 + 𝑔 𝑦𝑙 𝑧𝑙 = 𝐷𝑦𝑙 + 𝐸𝑣𝑙 + 𝑜𝑙

Use the periodic signal to separate the nonperiodic part (caused by the process noise and output noise) from the system nonlinearity If residual is nonstationary  process noise present

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Basic idea & principle

Measurement protocol using only one experiment

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Basic idea & principle

(proposed indicator)

 

2

ˆ t 

is proposed to be rearranged accordingly with the input

• f one period

Simulated example: NLSS

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 

0.5 1 0.5 0.4 0.3 , 0.2 , 1 0 , 0.3 0.8 0.6 0.3 A B C D                          

𝑔 𝑦𝑙 = tanh(𝑦𝑙(1)) ൞ 𝑦𝑙+1 = 𝐵𝑦𝑙 + 𝐶𝑣𝑙 + 𝑔 𝑦𝑙 + 𝑓𝑙 𝑧𝑙 = 𝐷𝑦𝑙 + 𝐸𝑣𝑙 + 𝑜𝑙

Simulated example: NLSS

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The process noise does pass through the static nonlinearity

Process Noise std: 0,01 Output Noise std: 0,01 P = 64

Simulated example: NLSS

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The process noise does NOT pass through the static nonlinearity

Process Noise std: 0,00 Output Noise std: 0,01 P = 64

Simulated example: NARMAX

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The process noise does pass through the static nonlinearity

Process Noise std: 0,10 Output Noise std: 0,01 P = 64

𝑧𝑙 = 𝑦𝑙 + 𝑜𝑙 𝑦𝑙 = 0,39𝑦𝑙−1 − 𝑣𝑙 + 0,1𝑦𝑙−1𝑣𝑙−1 + 𝑓𝑙

Simulated example: NARMAX

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The process noise does NOT pass through the static nonlinearity

Process Noise std: 0,00 Output Noise std: 0,01 P = 64

𝑧𝑙 = 𝑦𝑙 + 𝑜𝑙 𝑦𝑙 = 0,39𝑦𝑙−1 − 𝑣𝑙 + 0,1𝑦𝑙−1𝑣𝑙−1

Wiener-Hammerstein benchmark

Wiener-Hammerstein benchmark

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P = 16, Saturation nonlinearity

Nonlinearity Interpretation

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Taylor Approximation

𝑔 𝑦 + 𝜀 ≈ 𝑔 𝑦 + 𝑔′(𝑦)𝜀 std(𝑔 𝑦 + 𝜀 ) ≈ 𝑔′ 𝑦 std(𝜀)

Conclusion

Simple & Fast Applicable to a wide class of nonlinear systems Insight on the nonlinearity which the process noise goes through.

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