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Workshop on Nonlinear System Identification Benchmarks, April 2017

A methodology to compare two estimation methods for Parallel Hammerstein Models

Marc REBILLAT

DYSCO Group, PIMM, Arts et Métiers ParisTech, Paris, France

Maarten SCHOUKENS

ELEC Department, Vrije Universiteit Brussel, Belgium

Workshop on Nonlinear System Identification Benchmarks, April 2017

Parallel Hammerstein Models (PHMs)

Introduction

 Subclass of Volterra series: relatively general (to discuss)  Represented by N linear filters: rather simple

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Workshop on Nonlinear System Identification Benchmarks, April 2017

Introduction

 Various methods are available to estimate PHMs:

➢ LS: Least-square based methods ➢ ESS: Exponential sine sweep based methods ➢ …

How can we fairly compare them?

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Objective

Workshop on Nonlinear System Identification Benchmarks, April 2017

A problem that is linear in the parameters…

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I) LS method

 Output of a Parallel Hammerstein Model:

𝑧 𝑢 = ෍

𝑜=1 𝑂

෍

𝜐=0 𝑜ℎ

ℎ𝑜 𝜐 𝑣𝑜(𝑢 − 𝜐)

 An alternative way to write it: ➢ 𝑧 : Vector of output samples ➢ 𝐿 : Regressor matrix [input dependent] ➢ 𝜄 : Parameters to estimate [kernels ℎ𝑜(𝑢)]

𝑧 = 𝐿𝜄

 A least square solution:

෠ 𝜄 = 𝐿𝑈𝐿 −1𝐿𝑈𝑧

Workshop on Nonlinear System Identification Benchmarks, April 2017

Incorporating priori knowledge through regularization

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I) LS method

 What is known: ➢ The system is exponentially decaying (stability) ➢ Impulse responses samples are highly correlated (smoothness)  A regularized cost function: ➢ 𝑆 : Regularization matrix implemented as a block- diagonal matrix using the TC-kernel (Pillonetto, 2014)

𝑊

𝑠 =

𝑧 − 𝐿𝜄 2 + 𝜄𝑈𝑆𝜄

Workshop on Nonlinear System Identification Benchmarks, April 2017

Overview

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I) LS method

𝑧 = 𝐿𝜄 Regularization Least-square estimation  Direct Least-Squares (DLS): ➢ Input signal: Arbitrary (White Gaussian Noise [WGN] preferred) ➢ Parameters: FIR length, nonlinear order.  Regularized Least-Squares (RLS): ➢ Input signal: Arbitrary (WGN preferred) ➢ Parameters: FIR length, nonlinear order, 2 regularization hyperparameters.

Workshop on Nonlinear System Identification Benchmarks, April 2017

Effect of the inverse filter on the output signal

Output s(t) Inverse filter y(t)

Temporal alignment of the harmonics energy

∗ =

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II) ESS method

Workshop on Nonlinear System Identification Benchmarks, April 2017

 gn(t): contributions of the different harmonics

Temporal Windowing

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II) ESS method

Workshop on Nonlinear System Identification Benchmarks, April 2017

Linear relation ship between ℎ𝑙(𝑢) and 𝑕𝑜(𝑢):

g2(t): contribution of the 2nd harmonic h2(t): contribution of the power 2

cos2 𝑦 = 1 2 [1 + cos 2𝑦 ]

From harmonics to power… All the filters 𝒊𝒍(𝒖) are estimated.

ℎ1 𝑢 ⋮ ℎ𝑂(𝑢) = 𝐵 𝑕1 𝑢 ⋮ 𝑕𝑂(𝑢)

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II) ESS method

Workshop on Nonlinear System Identification Benchmarks, April 2017

Overview

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I) LS method

Inverse convolution Temporal windowing ℎ = 𝐵𝑕

 Exponential sine sweep method (ESS): ➢ Input signal: Exponential sine sweep (ESS) ➢ Parameters: Nonlinear order.  Non-parametric exponential sine sweep method (NP-ESS): ➢ Input signal: ESS ➢ Parameters: none.

Workshop on Nonlinear System Identification Benchmarks, April 2017

Performance indexes

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III) Comparison methodology

 What you pay: ➢ Computation time ➢ Parameters assignment  What you end up with: ➢ 𝑄𝐽1: Ability to reconstruct the output from an ESS ➢ 𝑄𝐽2: Ability to reconstruct the output from a WGN ➢ 𝑄𝐽3: Ability to estimate the Kernels: 𝑄𝐽3 = 1 𝑂 ෍

𝑜=1 𝑂

σ𝑙=0

𝐿

෠ ℎ𝑜 𝑢 − ℎ𝑜 𝑢

2

σ𝑙=0

𝐿

[ℎ𝑜 𝑢 ]2

Workshop on Nonlinear System Identification Benchmarks, April 2017

Experimental plan

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III) Comparison methodology

Workshop on Nonlinear System Identification Benchmarks, April 2017

PHM of order 𝑂 = 4

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IV) Application: system #1

Workshop on Nonlinear System Identification Benchmarks, April 2017

𝑄𝐽1: Ability to reconstruct the output of an ESS

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IV) Application: system #1

 ESS methods are computationally very efficient  LS methods are very precise

Workshop on Nonlinear System Identification Benchmarks, April 2017

𝑄𝐽2: Ability to reconstruct the output of a WGN

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IV) Application: system #1

 ESS methods can reconstruct WGN output  LS methods can be very precise

Workshop on Nonlinear System Identification Benchmarks, April 2017

𝑄𝐽3: Ability to estimate the Kernels

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IV) Application: system #1

 NP-ESS methods needs long enough signals  RLS methods more precise than DLS (specially with WGN)

Workshop on Nonlinear System Identification Benchmarks, April 2017

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IV) Application: system #2 [Bouc-Wen]

𝑄𝐽1: Ability to reconstruct the output of an ESS

 Large computational gap for equal performances  Large influence of input signal for LS methods

Workshop on Nonlinear System Identification Benchmarks, April 2017

𝑄𝐽2: Ability to reconstruct the output of a WGN

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IV) Application: system #2

 Large computational gap for almost equal performances  Large influence of input signal for LS methods

Workshop on Nonlinear System Identification Benchmarks, April 2017

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Conclusion

 ESS methods: ➢ Computationally very efficient ➢ Non parametric (or almost) ➢ Not extremely precise  LS methods: ➢ Computationally more intensive ➢ Can be extremely precise ➢ Parameters to be set

ESS methods could guide LS methods?

Overview

Workshop on Nonlinear System Identification Benchmarks, April 2017

Conclusion

But …

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 Performances of the Exponential Sine Sweep Method ➢ Managing and quantifying uncertainties… ➢ Extension to Parallel Wiener Models… ➢ Theoretical understanding of the estimator properties… ➢ Lower assumptions on input signal…

Workshop on Nonlinear System Identification Benchmarks, April 2017

Questions?

A methodology to compare two estimation methods for Parallel Hammerstein Models

Marc REBILLAT DYSCO Group, PIMM, Arts et Métiers ParisTech, Paris, France Maarten SCHOUKENS ELEC Department, Vrije Universiteit Brussel, Belgium