MEC 2016

FREQUENCY-DOMAIN SYSTEM IDENTIFICATION
FOR HIGH-PRECISION CONTROL DESIGN
Thomas Beauduin, Hiroshi Fujimoto (The University of Tokyo)

0-9010-1

06-Dec-1610010110210325 Slides

INTRODUCTION

1.1 Main References

TDI

FDI
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INTRODUCTION

1.2 System Identification for Control
System Modelling
Model Input

Physics-based Model

System Identification

Experiment Design

trajectory

Model Output
pos., vel. …

LMM, FEM, PDE, …

generator noise

Experimental Setup

Plant Input

process noise

Plant Output
pos., vel. …

excitation

Non-parametric Identification
measurement
noise

FRF Models

measurement
noise

Parametric Identification
Reference Input
trajectory

System Control
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Parametric Models

Modelled Output
pos., vel. …

Controller Model
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NON-PARAMETRIC

2.1 Overview of Non-Parametric Methods• Methods• Results

0000-50-50
1-100

coherence
-200

010

110SNR

1-100-200

0210

1

10-100-100-150-150

210210180

180180180909090

90

0000-90-90

101102-180-1801011022-5min experiment
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Two-inertia-system benchmark

210210210s experiment
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NON-PARAMETRIC

2.2 Discrete Fourier Transform – Problems• Aliasing errors: high frequency components mirrored at lower frequencies around Nyquist

aliasing• Leakage errors: signal spectrum smeared out due to finite measurement length
rectangular
window

leakage06-Dec-16

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NON-PARAMETRIC

2.2 Discrete Fourier Transform – Remedies• Anti-Aliasing: set high sampling frequencies, don’t excite near-Nyquist, anti-aliasing filters• Anti-Leakage: 2 solutions & 1 trade-off

noise level
noise (rec.
window)• Pitfalls:

long windows

averages necessary

remove transientsnoise level
noise (rec.
window)
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NON-PARAMETRIC

2.3 Periodic Excitation – Acquisition• Procedure

measure M consecutive periodsof steady state response
use a rectangular window• Data Obtained

Note :
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decrease FRF estimate uncertainty = M larger for given NT
increasing FRF frequency resolution = N larger for given NT

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NON-PARAMETRIC

2.3 Periodic Excitation – Processing• Frequency Response

Maximum likelihood approachassuming noise on both recordsfrequencyaveraging• VarianceSample variance of non-parametric systemmodel also called Cramer-Rao lower bound00-50-50

SNR-100.
non-parametric
noise model-100-150-150
102

180

18090

90

00-90

2-90-180

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NON-PARAMETRIC

2.4 Random Excitation – Acquisition• Procedure

Divide the input-output records
in M segments
use a Hanning window• Data Extraction

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NON-PARAMETRIC

2.4 Random Excitation – Processing• Frequency Response

Least squares approach assuming
input signal is exactly known• Coherence

Quantifies the quality of data by measuring
the input to output linear relationship001-100

1-100

coherence
-200

010

Note : coherences indicates presence of- extraneous noise- non-linear distortions- leakage errors- non-measured input or disturbance

10-200

02

10180

18090

90

00-90-90-180-180
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1

1

10

2

10

1

1

10

10

2

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NON-PARAMETRIC

2.5 Problems with Random Excitations• Longer experiment time

white noise input amplitude spectrum contains random large drops• Lower FRF resolution

windowing reduces leakage, but not in strong varying FRF frequencies, e.g. (anti-)resonances
leakage errors

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resulting coherence

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EXCITATION

3.1 Experiment Design• Constrained optimization
i.e. minimize piek variancee.g. max stage strokee.g. max actuator powere.g. embedded system• Non-parametric design criteria

minimal and flat SNR i.e. define two characteristics:
Crest factor

i.e. signal compactness

Time factor *i.e. efficiency of signal(*) required measurement time per frequency line to reach desired SNR06-Dec-16

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EXCITATION

3.2 Excitation Signals Comparison
Random noiseRandom

Binary sequence

22

110-1

Sweptsineor PRBS

221

1000-1-1-1-2-2
0

0.2

0.4

0.6

0.8

Periodic-2

0

1

0.2

0.4

0.6

0.8

1-2

020

20

10000Multisine

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

100.1

0.2

0.3

0.4

0.5

1000-100-200-20-10-20-300-40-20-40

00.1

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0.2

0.3

0.4

0.500.1

0.2

0.3

0.4

0.5

-400
0

0.1

0.2

0.3

0.4

0.5

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EXCITATION DESIGN

3.3 Multisine Optimization• Crest Factor optimization – Phase calculation
problemanalytical solution

numerical solution• Time Factor optimization – Amplitude spectrum shaping for flat SNR(*) example:
see beyond thecurrent loop
low-pass filter
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PARAMETRIC

4.1 Parametric Methods Overview

-20

10-40

0.5

-50
-60

0-100
-80

-0.5-1-150

-100
00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

110.5101102180180909000000.1

0.2

0.3

0.4

0.5

0.6

0.7

5.24s – SISO optimization
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Conventional

0.8

0.9

1102-180-180-1

1-90-90-0.5

101011021011020.32s – SIMO optimization
Proposed

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PARAMETRIC

4.2 Frequency Domain Identification (FDI)• Parametrization: Error-In-Variables structure-200-40-50-60• Optimization: non-linear least squares-100-80-150-10010102101801809090

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PARAMETRIC

4.2 FD Identification Algorithm• Problem Formulation: Non-linear Least Squares• Problem Reformulation

Approach 1: Analytical Linear Least Squares
biased, but usefulfor starting-values
Approach 2: Iteratively Linear Least Squaresa) SanathanKoerner:b) Gauss-Newton:c) Instrumental Variable:

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PARAMETRIC

4.2 FD Identification Weighting• General Formulation

Deterministic Approach

Stochastic Approachno a-priori information used
Weighted Least Squares (Markov) Estimator

Sample (co-)variances used
Maximum Likelihood Estimator

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PARAMETRIC

4.3 Time-Domain Identification (TDI)• Parametrization: parametric noise structure

PlantNoisemodeltransient• Model structures:

10.5

0-0.5

-1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

0.5

0-0.5

-1

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PARAMETRIC

4.3 Time-Domain Identification Methods• Estimation Framework
One-step-ahead prediction

One-step-ahead predictor outputPrediction errorPrediction error cost function

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5.1 HFLab Application Range

Machine-tool stages

Motors & converters

High-precision stages

In-Wheel-Motors

Robotics
http://hflab.k.u-tokyo.ac.jp/

other challenges …

Open Source Matlab Toolbox: www.github.com/thomasbeauduin/FdiTools

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EXPERIMENTS

5.2 High-precision Stage – Finite Element Model• Finite-Element-Analysis• Bode-diagrams

motor-side
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load-side
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EXPERIMENTS

5.3 Excitation Design – Multisine• Inverse Multisine design• Inner-loop design

221100-1-1

20-210-2-2-41102100310.5

00.05

0.1

0.15

0.2

0-20-20-40-40-60-60-80-80-901011021kHz – current control bandwidth
Weighted Least Square Estimation
Proposed Analyzer

10310010110210310010110210a) Quasi-Logarithmic gridwide spectrum – limited frequency linesb) Linear-Segmented grid
limited spectrum – high precision data06-Dec-16

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3

EXPERIMENTS

5.4 Non-parametric identification – SIMO
• Maximum likelihood estimator (MLE).
Drive – side0Load – side0-50-50dataset 1-100

10-1100

set 2101102

dataset 1-100

10310180180909000-90-90-180-180-270-1100

set 2101102103-270
10-1

100

10110210310-1

100

1011021032 x 10s – segmented experiments
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Proposed Analyzer

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EXPERIMENTS

5.5 Parametric Identification – SIMO
• Maximum likelihood estimator (MLE)

Drive – side

Load – side

00-90-90

10-110010110210310200

200

100100-110010110210300-100-100-200-200-300-300
10-106-Dec-16

10010110210310-1

5.24s – SIMO optimization
Proposed Analyzer

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MEC 2016

FREQUENCY-DOMAIN SYSTEM IDENTIFICATION
FOR HIGH-PRECISION CONTROL DESIGN
Thomas Beauduin, Hiroshi Fujimoto (The University of Tokyo)

ご清聴ありがとうございました
Thank you for your kind attention

06-Dec-1625 Slides

C.

CONCLUSIONSFrequency Domain Identification for Control• STEP 1: Excitation Design• STEP 2: Non-Parametric Identification• STEP 3: Parametric Identification

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