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1 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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2 Uncertainty bounding in PE identification 3.For a given single realization determine a set of for which, within a probability level of holds that Delft Center for Systems and Control 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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3 Delft Center for Systems and Control probability region for

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4 Delft Center for Systems and Control 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 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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5 Contents Classical OE result New OE approach Extension towards finite time (?) Options for extending to Summary Delft Center for Systems and Control

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6 OE modelling (classical) 1-step ahead predictor: Identification criterion: Based on Taylor approximation of Conditioned on asymp. norm. of

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7 Delft Center for Systems and Control 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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8 Model uncertainty bounding requires: (asymptotic) normality of with covariance Q Replacement of by an estimate leading to ellipsoid determined by Delft Center for Systems and Control Alternative Same as before but conditions are relaxed

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9 Delft Center for Systems and Control Conclusion Classical results with approximated by sample estimates, requires to become (asymptotically) Gaussian rather than Benefit: relaxation of conditions for normality

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10 Delft Center for Systems and Control 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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11 Delft Center for Systems and Control Corresponding uncertainty Only problem: condition of independent and is not 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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12 Delft Center for Systems and Control 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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13 Delft Center for Systems and Control Component related to numerator parameter:

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14 Delft Center for Systems and Control Component related to denominator parameter:

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15 Delft Center for Systems and Control Conclusion New analysis arrives at correct Gaussian distribution for uncertainty quantification, for a remarkably small number of N

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16 Delft Center for Systems and Control Options for extending to In analysis, replace by becomes: output noiseunmodelled dynamics Results to be quantified in frequency domain by using a linearized mapping

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17 Delft Center for Systems and Control 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) 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

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1 Although they are biased in finite samples if Part (2) of Assumption C.7 is violated, OLS estimators are consistent if Part (1) is valid. We will demonstrate.

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