1 probabilistic uncertainty bounding in output error models with unmodelled dynamics 2006 american...

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Probabilistic Uncertainty Bounding in Output Error Models with Unmodelled Dynamics

2006 American Control Conference, 14-16 June 2006, Minneapolis, Minnesota

Delft Center for Systems and Control

Sippe Douma and Paul Van den Hof

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Uncertainty bounding in PE identification

3. For a given single realization determine a set offor which, within a probability level of holds that


1. Analyse statistical properties of estimator(mapping data to parameter) through its pdf

2. Under (asymptotic) assumptions and no bias

with an analytical expression for covariance matrix P

4. This set is identical to the set of that forms an probability set of the pdf

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probability region for

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• Approximations to be made:• Employing asymptotic Gaussian distribution

of pdf• Assumption (no bias)• Obtaining P through Taylor approximation

(OE/BJ)• Replacing covariance matrix P by estimateThe

message • Estimator statistics are not necessary for obtaining probabilistic parameter uncertainty regions;There are alternatives with attractive properties

(Douma, Van den Hof, CDC/ECC-05)

Here:• Applied to Output Error models• Validity for finite-time• Options for

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Contents

• Classical OE result• New OE approach• Extension towards finite time (?)• Options for extending to • Summary


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OE modelling (classical)

1-step ahead predictor:

Identification criterion:

Based on Taylor approximation of

Conditioned on asymp. norm. of

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OE modelling (alternative)

Taylor approximation of :

which when substituted above, leads to

= linear regression type of expression for parameter error

known

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Model uncertainty bounding requires:

• (asymptotic) normality of with covariance Q• Replacement of

by an estimate

leading to ellipsoid determined by


Alternative

Same as before but conditions are relaxed

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ConclusionClassical results with approximated by sample estimates, requires

to become (asymptotically) Gaussian rather than

Benefit: relaxation of conditions for normality

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One step further (towards finite-time)

With svd: it follows that

Lemma:

If unitary and random, and e Gaussian with cov(e)=2 I,and and e independent, then is Gaussian with Cov = 2 I.

This would suggest that is Gaussian for any value of N.

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Corresponding uncertainty

Only problem: condition of independent and isnot satisfied, since is a function of

Starting from:

It follows that (for finite N):

with

Only needs to be replaced by an estimated expression

However: this does not appear devastating in practice!

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Simulation example:

First order OE system:

identified with 1st order OE model.

Compare empirical distributions of

and

for different values of N, on the basis of 5000 Monte Carlo simulations

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Component related to numerator parameter:

-50 0 50

(T

)-1

T e

-2 0 2

N = 2

VTe

-2 0 2

-2 0 2

N = 5

-0.5 0 0.5

-2 0 2

N = 25

-0.2 0 0.2

-2 0 2

N = 50

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Component related to denominator parameter:

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ConclusionNew analysis arrives at correct Gaussian distribution for uncertainty quantification, for a remarkably small number of N

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Options for extending to In analysis, replace by

becomes:

output noise unmodelled dynamics

Results to be quantified in frequency domain by usinga linearized mapping

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Summary

• There are alternatives for parameter uncertaintybounding, without constructing pdf of estimator

• Applicable to ARX, OE and also BJ models (see also Douma & VdHof, CDC/ECC-2005, SYSID2006)

• And with good options to be extended to the situation of approximate models

• Leading to simpler and less approximative expressions, remarkably robust w.r.t. finite time properties

1 probabilistic uncertainty bounding in output error models with unmodelled dynamics 2006 american...

Documents

control slide

control oe

minnesota delft center

summary delft center

pdf slide

normality slide

control alternative

relaxed slide