The Local Rational Method for Norm EstimationEgon GeerardynVUB-ELEC, Tom OomenTU/e-CST, Johan SchoukensVUB-ELEC

egon.geerardyn@vub.ac.be

This work is supported by the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (Methusalem), by the Belgian Government through the Interuniversity Poles of Attraction (IAP VII-Dysco) Program, and by the Innovational Research Incentives Scheme under the VENI grant “Precision Motion: Beyond the Nanometer” (no. 13073) awarded by NWO (Netherlands Organization for Scientific Research) and STW (Dutch Science Foundation).

ӧ controller: simple P̂ ӧ unmodeled dynamics in ӧ may contain sharp peaks ӧ estimate & compute kk1 ӧ robust controller design

Problem Situation

Local Rational Method (LRM) & interpolation

RefeRences

R. Pintelon, J. Schoukens, G. Vandersteen, K. Barbé, Estimation of nonparametric noise and FRF models for multivariable systems, Mechanical Systems and Signal Processing, 2010, Volume 24, Issue 3, Pages 573-616, http://dx.dor.org/10.1016/j.ymssp.2009.08.009.T. McKelvey, G. Guérin, Non-Parametric Frequency Response Estimation Using a Local Rational Model, Proceedings of the 16th IFAC Symposium on System Identification, July 11-13, 2012, Brussels, Belgium.E. Geerardyn, T. Oomen, J. Schoukens, Enhancing H∞ Norm Estimation using Local LPM/LRM Modeling: Applied to an AVIS, Proceedings of the 19th IFAC World Congress, August 24-29, 2014, Cape Town, South Africa.D. de Vries, P. van den Hof, Quantification of Uncertainty in Transfer Function Estimation: a Mixed Probabilistic-Worst-Case Approach, Automatica, 1995, Volume 31, Number 3, Pages 543-547, http://dx.doi.org/10.1016/0005-1098(95)98483-M.T. Oomen, O. Bosgra, System identification for achieving robust performance, Automatica, 2012, Volume 48, Issue 9, Pages 1975-1987, http://dx.doi.org/10.1016/j.automatica.2012.06.011.

Results for an Active Vibration Isolation System (AVIS)

310 320 328 340 350

0

2.5

10

20

30

r1 u

r2 C P ye uc

−

complex plant

A (!k)

ӧ Vastly improved kk1 using LRM ӧ Short measurement time ӧ Almost for free ӧ Robust controller with guarantees ӧ MIMO extension

Y (!k) = B (!k)U (!k) + T (!k) + noise

smooth over !

Fit A, B, T around each bin !k:

model model error

(!k

+δ!)

(!k+1 +

δ!)

||

Frequency

||

Frequency resolution is key for determining kk1

0

10

20

30

40

50

f = 1

T

kk1 ⇡ 39

kk1 = 50

Frequency [Hz]

Amplitude|

|

Egon Geerardyn e.a. LPM for H1 8 / 17

Frequency

FRF-based: optimistic

Actual value

∆(ωk + δω) ≈ B(ωk + δω)

A(ωk + δω)=

NB∑

i=0

bi(k)(δω)i

1 +

NA∑

i=1

ai(k)(δω)i

||

7.5 dB improvement

LRM

[dB]