Transcript Document

Probabilistic Uncertainty Bounding in
Output Error Models with Unmodelled
Dynamics
Sippe Douma and Paul Van den Hof
2006 American Control Conference, 14-16 June 2006, Minneapolis, Minnesota
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Uncertainty bounding in PE identification
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
3. For a given single realization
determine a set of
for which, within a probability level of %, holds that
4. This set is identical to the set of
% probability set of the pdf
that forms an
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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 estimate
The message (Douma, Van den Hof, CDC/ECC-05)
• Estimator statistics are not necessary for obtaining
probabilistic parameter uncertainty regions;
There are alternatives with attractive properties
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
known
= linear regression type of expression for parameter error
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Alternative
Model uncertainty bounding requires:
• (asymptotic) normality of
• Replacement of
with covariance Q
by an estimate
leading to ellipsoid determined by
Same as before but conditions are relaxed
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Conclusion
Classical results with
estimates, requires
approximated by sample
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)=s2 I,
and
and e independent, then
is Gaussian with
Cov = s2 I.
This would suggest that
is Gaussian for any value of N.
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Corresponding uncertainty
Starting from:
It follows that (for finite N):
with
Only
needs to be replaced by an estimated expression
Only problem: condition of independent
and
is
not satisfied, since
is a function of
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:
N=2
N = 25
N = 50
T
-1
T
(  )  e
N=5
0
50
-2
0
2
-2
0
2
-0.5
0
0.5
-0.2
0
0.2
-2
0
2
T
V e
-50
-2
0
2
-2
0
2
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Component related to denominator parameter:
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Conclusion
New 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 using
a linearized mapping
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Summary
• There are alternatives for parameter uncertainty
bounding, without constructing pdf of estimator
• Applicable to ARX, OE and also BJ models
(see also Douma & VdHof, CDC/ECC-2005, SYSID2006)
• Leading to simpler and less approximative expressions,
remarkably robust w.r.t. finite time properties
• And with good options to be extended to the situation of
approximate models
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